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FAT CHANCE

In a world where we are constantly being asked to make decisions based

on incomplete information, facility with basic probability is an essential
skill. This book provides a solid foundation in basic probability theory
designed for intellectually curious readers and those new to the subject.
Through its conversational tone and careful pacing of mathematical
development, the book balances a charming style with informative
discussion.
This text will immerse the reader in a mathematical view of the
world, giving them a glimpse into what attracts mathematicians to the
subject in the first place. Rather than simply writing out and mem-
orizing formulas, the reader will come out with an understanding of
what those formulas mean, and how and when to use them. Readers
will also encounter settings where probabilistic reasoning does not
apply or where our intuition can be misleading. This book establishes
simple principles of counting collections and sequences of alternatives
and elaborates on these techniques to solve real-world problems both
inside and outside the casino. Readers at any level are equipped to
consider probability at large and work through exercises on their own.

Benedict Gross is Leverett Professor of Mathematics, Emeritus at

Harvard University and Professor of Mathematics at UC San Diego. He
has taught mathematics at all levels at Princeton, Brown, Harvard, and
UCSD, and served as the Dean of Harvard College from 2003–2007.
He is a member of the American Academy of Arts and Sciences and
the National Academy of Science. Among his awards and honors are the
Cole Prize from the American Mathematical Society and a MacArthur
Fellowship. His research is primarily in number theory.

Joe Harris is the Higgins Professor of Mathematics at Harvard Univer-

sity. He has been at Harvard since 1988 and was previously on the
faculty at MIT and Brown. He is a member of the American Academy of
Arts and Sciences and the National Academy of Science. Throughout his
career, he has been deeply committed to education at every level, which
led to a partnership with Benedict Gross to develop the Harvard course
“Fat Chance,” the inspiration for the book of the same title. He is author
of several books including 3264 and All That, Algebraic Geometry, and
The Geometry of Schemes.

Emily Riehl is an Assistant Professor of Mathematics at Johns Hopkins

University and previously was a Benjamin Peirce and NSF postdoctoral
fellow at Harvard University. She has published over twenty papers and
written two books: Categorical Homotopy Theory and Category Theory
in Context. She has been awarded an NSF grant and a CAREER award to
support her work and has been recognized for excellence in teaching at
both Johns Hopkins and Harvard.
Fat Chance
Probability from 0 to 1

BENEDICT GROSS
Harvard University

JOE HARRIS
Harvard University

EMILY RIEHL
Johns Hopkins University
University Printing House, Cambridge CB2 8BS, United Kingdom

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Cambridge University Press is part of the University of Cambridge.

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education, learning, and research at the highest international levels of excellence.

www.cambridge.org
Information on this title: www.cambridge.org/9781108482967
DOI: 10.1017/9781108610278

© Benedict Gross, Joe Harris, and Emily Riehl 2019

This publication is in copyright. Subject to statutory exception

and to the provisions of relevant collective licensing agreements,
no reproduction of any part may take place without the written
permission of Cambridge University Press.

First published 2019

Printed in the United Kingdom by TJ International Ltd., Padstow, Cornwall

A catalogue record for this publication is available from the British Library.

Library of Congress Cataloging-in-Publication Data

Names: Gross, Benedict H., 1950- author. | Harris, Joe, 1951– author. | Riehl, Emily, author.
Title: Fat chance : probability from 0 to 1 / Benedict Gross (Harvard University, Massachusetts),
Joe Harris (Harvard University, Massachusetts), Emily Riehl (Johns Hopkins University).
Description: Cambridge ; New York, NY : Cambridge University Press, c2019. |
Includes bibliographical references and index.
Identifiers: LCCN 2018058461| ISBN 9781108482967 (hardback : alk. paper) |
ISBN 9781108728188 (pbk. : alk. paper)
Subjects: LCSH: Probabilities–Popular works. | Probabilities–Problems, exercises, etc.
Classification: LCC QA273.15 .G76 2019 | DDC 519.2–dc23
LC record available at https://lccn.loc.gov/2018058461
ISBN 978-1-108-48296-7 Hardback
ISBN 978-1-108-72818-8 Paperback
Cambridge University Press has no responsibility for the persistence or accuracy
of URLs for external or third-party internet websites referred to in this publication
and does not guarantee that any content on such websites is, or will remain,
accurate or appropriate.
Contents

Preface page ix

Part I. Counting
1 Simple counting . . . . . . . . . . . . . . . . . . . 3
1.1 Counting numbers 3
1.2 Counting divisible numbers 5
1.3 “I’ve reduced it to a solved problem.” 6
1.4 Really big numbers 7
1.5 It could be worse 9

2 The multiplication principle . . . . . . . . . . . . . 11

2.1 Choices 11
2.2 Counting words 12
2.3 A sequences of choices 15
2.4 Factorials 16
2.5 When order matters 18

3 The subtraction principle . . . . . . . . . . . . . . 21

3.1 Counting the complement 21
3.2 The art of counting 23
3.3 Multiple subtractions 26

4 Collections . . . . . . . . . . . . . . . . . . . . 30
4.1 Collections vs. sequences 30
4.2 Binomial coefficients 31
4.3 Counting collections 36
4.4 Multinomials 42
4.5 Something’s missing 45

5 Games of chance . . . . . . . . . . . . . . . . . . 47
5.1 Flipping coins 47
5.2 Rolling dice 50
5.3 Playing poker 53
5.4 Really playing poker 57
5.5 Bridge 58
5.6 The birthday problem 60

v
vi Contents

Interlude
6 Pascal’s triangle and the binomial theorem . . . . . . 65
6.1 Pascal’s triangle 65
6.2 Patterns in Pascal’s triangle 69
6.3 The binomial theorem 72

7 Advanced counting . . . . . . . . . . . . . . . . . 75
7.1 Collections with repetitions 75
7.2 Catalan numbers 80
7.3 A recursion relation 81
7.4 Another interpretation 83
7.5 The closed formula 85
7.6 The derivation 86
7.7 Why do we do these things, anyway? 90

Part II. Probability

8 Expected value . . . . . . . . . . . . . . . . . . . 95
8.1 Chuck-A-Luck 95
8.2 Why we’re spending so much time at the casino 100
8.3 Expected value 101
8.4 Strategizing 103
8.5 Medical decision-making 107

9 Conditional probability . . . . . . . . . . . . . . 109

9.1 The Monty Hall problem 109
9.2 Conditional probability 111
9.3 Independence 115
9.4 An election 118
9.5 Bayes’ theorem 121
9.6 The zombie apocalypse 125
9.7 Finally, Texas Hold ’Em 127

10 Life’s like that: unfair coins and loaded dice . . . . . . 132

10.1 Unfair coins 132
10.2 Bernoulli trials 136
10.3 Gambler’s ruin 140

11 Geometric probability . . . . . . . . . . . . . . . 146

11.1 Coin tossing at the carnival 146
11.2 At the dining hall 148
11.3 How to tell if you’re being stalked 152
11.4 Queuing theory 153

Part III. Probability at large

12 Games and their payoffs . . . . . . . . . . . . . . 159
12.1 Games and variance 160
12.2 Changing the payoffs 163
Contents vii

12.3 The normalized form of a game 165

12.4 Adding games 167

13 The normal distribution . . . . . . . . . . . . . . 171

13.1 Graphical representation of a game 171
13.2 Every game is normal 178
13.3 The importance of the standard deviation 181
13.4 Polling 183
13.5 The fine print on polling 185
13.6 Mixing games 185

14 Don’t try this at home . . . . . . . . . . . . . . . 187

14.1 Inverting conditional probabilities 187
14.2 False positives 188
14.3 Abuse of averages 189
14.4 Random correlation 190
14.5 What we think is random, isn’t; what we think isn’t
random, is 191

A Boxed formulas . . . . . . . . . . . . . . . . . . 194

B Normal table . . . . . . . . . . . . . . . . . . . 197

Index 199
Preface

Suppose a friend comes up to you in a bar with the following chal-

lenge. He asks you for a quarter—which you hand over somewhat
reluctantly—and tells you he’s going to flip it six times and record the
outcome: heads or tails. Your friend then offers you the following bet: if
you can correctly predict the number of heads, he’ll buy the next round
of drinks. But if the outcome is different from what you guessed, the
next round is on you. On each individual flip heads and tails are equally
likely. So it seems that three heads and three tails is the most likely
outcome. But is this a good bet?
Or, you and a friend are driving to the movies and find a place to
park a mile away from the theater. It’s a busy area, but you know there’s
some chance that if you drive on, you’ll find a space that’s only half a
mile away. It’s also possible that if you look for a better spot, you’ll have
to come back to this one, which might no longer be there. How do you
decide what to do? And how does it affect your decision if the movie
starts in 25 minutes?
In a school election, a vast majority of the left-handed voters prefer
Tracy, who has advocated strongly for more left-handed desks to be put
in classrooms. Does this mean that Tracy is likely to win? And while
we’re at it, what does a “3% margin of error” in a poll really mean?
Or suppose you go to the doctor with your uncle and are told that
his level of prostate-specific antigen (PSA) is high. The doctor reports
that a large proportion of people with prostate cancer have elevated
PSA levels. How worried should you be?
Finally, suppose you want to gamble with your life’s savings, totaling
$1,000, while playing roulette. Is it better to wager it all on a single bet,
or to bet just $1 at a time until you either double your money or go
bankrupt?
At its heart, probability is about decision-making, at least in contexts
when it’s possible to determine how mathematically likely a given event
would be. In this book we embark upon a journey that will allow us
to discover the answers to all of the questions posed above, using the
mathematics of probability theory to help us identify the ways in which
your intuition may lead you astray.
We’ll discuss the gambler’s fallacy and its opposite, the hot hand
fallacy in basketball, and try to understand why random correlations
arise in large data sets, such as the “coincidence” that means that it’s
likely that two people in a room of 30 share a birthday. We’ll explore
ix
x Preface

the mathematical distinction between events that are independent,

in the sense of being unrelated, and those that are “positively” or
“negatively” correlated. We’ll look at games like slots, poker, and
roulette to understand how Las Vegas stays in business, even when
it’s possible to accurately compute the expected value of each game.
And we try to intersperse more serious discussions, involving medical
decision-making and the zombie apocalypse, with some lighthearted
questions we explore just for fun.
Fat Chance began as a course in the program of General Education at
Harvard College. Two of us (Dick and Joe) taught it to help students who
were not majoring in mathematics or the sciences develop some basic
skills in counting and probability. We also hoped to use the course to
describe a mathematical view of the world, and to show students what
attracted us to math in the first place.
After we gave the course in the classroom a few times, we were
approached by HarvardX to develop lectures and problems, in order to
present the course online. This book is intended both as a companion
to our online course and for the general reader who wants to learn the
basics of probability.
Perhaps somewhat counterintuitively, we think of Fat Chance as akin
to an introductory language course. The heart of a language course is
not the memorization of a lot of vocabulary and verb tenses—though
invariably there is a lot of that involved—but rather the experience of
thinking and speaking in a different tongue. In the same way, in this
book there are of necessity a fair number of techniques to learn and
calculations to carry out. But they are just a means to an end: our goal
is to give you the experience of thinking mathematically, and the ability
to calculate, or at least estimate, probabilities that come up in many
real-world situations.
One way to try to learn a new language is to watch a lot of foreign
films with the subtitles turned off. Eventually you’ll learn to speak a
few phrases, but, appealing as this method of studying may sound, it’s
not exactly the most efficient tactic to achieve fluency. A better strategy
would be to find a few native speakers and attempt to have a conversa-
tion. You’re bound to say the wrong thing on a few occasions, which can
be embarrassing, but the evidence suggests you’ll learn much faster this
way, as much because of your mistakes as despite them. For this reason,
we’ve included some challenge exercises for the reader to puzzle over
throughout the text in hopes that many of you will take this opportu-
nity to try out for yourselves the probabilistic reasoning techniques we
introduce.
What sort of prerequisites does this book have? Well, the technical
answer to that is “very few.” Some familiarity with the topics in a high
school algebra course (manipulation of fractions, comfort with the
use of letters to stand for numbers) should be plenty. Probably more
important is a less quantifiable requirement: we ask the reader to be
prepared to approach the mathematics with a spirit of adventure and
exploration, with the understanding that, while some work will be
required, the experience at the end will be well worth it.
Preface xi

We are indebted to a great number of people who helped us create

this book. Our treatment of the topics in the first part of the book
was clearly influenced by Ivan Niven in his book, Mathematics of
Choice. Many graduate students at Harvard served as teaching fellows
when we gave the course in the classroom, and they helped to shape
both the material and the exercises. Special thanks go to Andrew
Rawson and all the videographers and editors in the HarvardX studios,
who made the filming of Fat Chance such a pleasure, to Cameron
Krulewski, who helped teach the online course and gave useful hands-
on feedback about the exercises, and to Devlin Mallory, who helped us
with copyediting and figure formatting. We are indebted to our editor,
Katie Leach, at Cambridge University Press, for her patience and her
many helpful suggestions on the possible audience for this book.
Now, let’s get down to work!
PA R T I

Counting
 1 Simple counting

It’s hard to begin a math book. A few chapters in, it gets easier: by then, writer and
reader have—or think they have—a common sense of the level of the book, its pace,
language, and goals; at that point, communication naturally flows more smoothly. But
getting started is awkward.
As a consequence, it’s standard practice in math textbooks to include a throwaway
chapter or two at the beginning. These function as a warmup before we get around
to the part of the workout that involves the heavy lifting. An introductory chapter
typically has little or no technical content, but rather is put there in the hope of estab-
lishing basic terminology and notation, and getting the reader used to the style of the
book, before launching into the actual material. Unfortunately, the effect may be the
opposite: a chapter full of seemingly obvious statements, expressed in vague language,
can have the effect of making the reader generally uneasy without actually conveying
any useful information.
Well, far be it from us to deviate from standard practice! The following is our intro-
ductory chapter. But here’s the deal: you can skip it if you find the material too easy
(unlike the case of power lifting, where a thorough warmup is absolutely necessary).
Really. Just go right ahead to Chapter 2 and start there.

1.1 COUNTING NUMBERS

To start things off, we’d like to talk about counting, because that’s how numbers first
entered our world. It was four or five thousand years ago that people first developed
the concept of numbers, probably in order to quantify their possessions and make
transactions—my three pigs for your two cows and the like. And the remarkable thing
that people discovered about numbers is that the same system of numbers—1, 2, 3, 4,
and so on—could be used to count anything: beads, bushels of grain, people living in
a village, forces in an opposing army. Numbers can count anything: numbers can even
count numbers.
And that’s where we’ll start. The first problem we’re going to pose is simply: how
many numbers are there between 1 and 10?
At this point you may be wondering if it’s too late to get your money back for this
book. Bear with us! We’ll get to stuff you don’t know soon enough. In the meantime,
write them out and count:

1, 2, 3, 4, 5, 6, 7, 8, 9, 10;
3
4 Chapter 1. Simple counting

there are 10. How about between 1 and 11? Well, that’s one more, so there are 11.
Between 1 and 12? 12, of course.
Well, that seems pretty clear, and if we now asked you, for example, how many
numbers there are between 1 and 57, you wouldn’t actually have to write them out
and count; you’d figure (correctly) that the answer would be 57.
OK, then, let’s ramp it up a notch. Suppose we ask now: how many numbers are
there between 28 and 83, inclusive? (“Inclusive” means that, as before, we include
both 28 and 83 in the count.) Well, you could do this by making a list of the numbers
between 28 and 83 and counting them, but you have to believe there’s a better way
than that.
Here’s one: suppose you did write out all the numbers between 28 and 83:

28, 29, 30, 31, 32, ......, 82, 83.

(Here the dots invite the reader to imagine that we’ve written all the numbers in
between in an unbroken sequence. We’ll use this convention when it’s not possible
or desirable to write out a sequence of numbers in full.) Now subtract the number 27
from each of them. The list now starts at 1, and continues up to 83 − 27 = 56:

1, 2, 3, 4, 5, ......, 55, 56.

From what we just saw we know there are 56 numbers on this list; so there were 56
numbers on our original list as well.
It’s pretty clear also that we could do this to count any string of numbers. For exam-
ple, if we asked how many numbers there are between 327 and 573, you could similarly
imagine the numbers all written out:

327, 328, 329, 330, 331, ......, 572, 573.

Next, subtract the number 326 from each of them; we get the list

1, 2, 3, 4, 5, ......, 246, 247,

and so we conclude that there were 573 − 326 = 247 numbers on our original list.
Now, there’s no need to go through this process every time. It makes more sense
to do it once with letters standing for arbitrary numbers, and in that way work out a
formula that we can use every time we have such a problem. So, imagine that we’re
given two whole numbers n and k, with n the larger of the two, and we’re asked: how
many numbers are there between k and n, inclusive?
We do this just the same way: imagine that we’ve written out the numbers from k to
n in a list
k, k + 1, k + 2, k + 3, . . . . . . , n − 1, n
and subtract the number k − 1 from each of them to arrive at the list

1, 2, 3, 4, ......, n − 1 − (k − 1), n − (k − 1).

Now we know how many numbers are on the list: it’s n − (k − 1) or, more simply,
n − k + 1.1 Our conclusion, then, is that:
1 Is it obvious that n − (k − 1) is the same as n − k + 1? If not, take a moment out to convince yourself:
subtracting k − 1 is the same as subtracting k and then adding 1 back. In this book we’ll usually
carry out operations like this without comment, but you should take the time to satisfy yourself
that they make sense.
 1.2. Counting divisible numbers 5

The number of whole numbers between k and n inclusive is n − k + 1.

So, for example, if someone asked “How many numbers are there between 342 and
576?” we wouldn’t have to think it through from scratch: the answer is 576 − 342 + 1,
or 235.
Since this is our first formula, it may be time to bring up the whole issue of the role
of formulas in math. As we said, the whole point of having a formula like this is that we
shouldn’t have to recreate the entire argument we used in the concrete examples above
every time we want to solve a similar problem. On the other hand, it’s also important to
keep some understanding of the process, and not to treat the formula as a “black box”
that spews out answers (regardless of the black box we drew to call attention to the
general formula). Knowing how the formula was arrived at helps us to know both when
it’s applicable, and how it can be modified to deal with other situations when it’s not.
Exercise 1.1.1. How many whole numbers are there between 242 and 783?
Exercise 1.1.2. The collection of whole numbers, together with their negatives and the
number zero, form a number system called the integers.
1. Suppose n and k are both negative numbers. How many negative numbers are there
between k and n inclusive?
2. Suppose n is positive and k is negative. How many integers are there between k and
n inclusive?

1.2 COUNTING DIVISIBLE NUMBERS

Now that we’ve done that, let’s try a slightly different problem: suppose we ask “How
many even numbers are there between 46 and 104?”
In fact, we can approach this the same way: imagine that we did make a list of all
even numbers, starting with 46 and ending with 104:

46, 48, 50, 52, ......, 102, 104.

Now, we’ve just learned how to count numbers in an unbroken sequence. And we can
convert this list to just such a sequence if we just divide all the numbers on the list by
2: doing that, we get the sequence

23, 24, 25, 26, ......, 51, 52

of all whole numbers between 46/2, or 23, and 104/2, or 52. Now, we know by the formula
we just worked out how many numbers there are on that list: there are

52 − 23 + 1 = 30

numbers between 23 and 52, so we conclude that there are 30 even numbers between
46 and 104.
One more example of this type: let’s ask the question, “How many numbers between
50 and 218 are divisible by 3?” Once more we use the same approach: imagine that we
made a list of all such numbers. But notice that 50 isn’t the first such number, since 3
 6 Chapter 1. Simple counting

doesn’t divide 50 evenly: in fact, the smallest number on our list that is divisible by 3 is
51 = 3 × 17. Likewise, the last number on our list is 218, which isn’t divisible by 3. The
largest number on our list which is divisible by 3 is 216, which is: 3 × 72 = 216. So the
list of numbers divisible by 3 would look like

51, 54, 57, 60, ......, 213, 216.

Now we can do as we did before, and divide each number on this list by 3. We arrive at
the list
17, 18, 19, 20, . . . . . . , 71, 72
of all whole numbers between 17 and 72, and there are

72 − 17 + 1 = 56

such numbers.
Now it’s time to stop reading for a moment and do some yourself:
Exercise 1.2.1.
1. How many numbers between 242 and 783 are divisible by 6?
2. How many numbers between 17 and 783 are divisible by 6?
3. How many numbers between 45 and 93 are divisible by 4?
Exercise 1.2.2.
1. In a sports stadium with numbered seats, every seat is occupied except seats 33
through 97. How many seats are still available?
2. Suppose the fans are superstitious and only want to sit in even-numbered seats
because otherwise they fear that their team will lose. How many even-numbered
seats are still available in the block of seats numbered 33 through 97?
Exercise 1.2.3. In a non-leap year of 365 days starting on Sunday, January 1st, how
many Sundays will there be? How many Mondays will there be?

1.3 “I’VE REDUCED IT TO A SOLVED PROBLEM.”

Note one thing about the sequence of problems we’ve just done. We started with
a pretty mindless one—the number of numbers between 1 and n—which we could
answer more or less by direct examination. The next problem we took up—the number
of numbers between k and n—we solved by shifting all the numbers down to whole
numbers between 1 and n − k + 1. In effect we reduced it to the first problem, whose
answer we knew. Finally, when we asked how many numbers between two numbers
were divisible by a third, we answered the question by dividing all the numbers, to
reduce the problem to counting numbers between k and n.
This approach—building up our capacity to solve problems by reducing new prob-
lems to ones we’ve already solved—is absolutely characteristic of mathematics. We
start out slowly, and gradually accumulate a body of knowledge and techniques; the
goal is not necessarily to solve each problem directly, but to reduce it to a previously
solved problem.
There’s even a standard joke about this:
 1.4. Really big numbers 7

A mathematician walks into a room. In one corner, she sees an empty bucket.
In a second corner, she sees a sink with a water faucet. And, in a third corner,
she sees a pile of papers on fire. She leaps into action: she picks up the bucket,
fills it up at the faucet, and promptly douses the fire.
The next day, the same mathematician returns to the room. Once more,
she sees a fire in the third corner, but this time sitting next to it there’s
a full bucket of water. Once more she leaps into action: she picks up the
bucket, drains it into the sink, places it empty in the first corner and leaves,
announcing: “I’ve reduced it to a previously solved problem!”

Well, maybe you had to be there. But there is a real point to be made here. It’s simply
this: the ideas and techniques developed in this book are cumulative, each one resting
on the foundation of the ones that have come before. We’ll occasionally go off on
tangents and pursue ideas that won’t be used in what follows, and we’ll try to tell you
when that occurs. But for the most part, you need to keep up: that is, you need to work
with the ideas and techniques in each section until you feel genuinely comfortable
with them, before you go on to the next.
It’s worth remarking also that the cumulative nature of mathematics in some ways
sets it apart from other fields of science. The theories of physics, chemistry, biology,
and medicine we subscribe to today flatly contradict those held in the seventeenth and
eighteenth centuries—it’s fair to say that medical texts dealing with the proper appli-
cation of leeches are of interest primarily to historians, and we’d bet your high school
chemistry course didn’t cover phlogiston.2 By contrast, the mathematics developed at
that time is the cornerstone of what we’re doing today.
Exercise 1.3.1. How many numbers between 242 and 783 are not divisible by 6?
Exercise 1.3.2. A radio station mistakenly promises to give away two concert tickets to
every thirteenth caller as opposed to offering two concert tickets only to the thirteenth
caller. They receive 428 calls before the station manager realizes the mistake. How
many concert tickets has the radio station promised to give away?

1.4 REALLY BIG NUMBERS

As long as we’re talking about the origins of numbers, let’s talk about another impor-
tant early development: the capacity to write down really big numbers. Think about it:
once you’ve developed the concept of numbers, the next step is to figure out a way to
write them down. Of course, you can just make up an arbitrary new symbol for each
new number, but this is inherently limited: you can’t express large numbers without a
cumbersome dictionary.
One of the first treatises ever written on the subject of numbers and counting was
by Archimedes, who lived in Syracuse (part of what was then the Greek empire) in
the third century BC. The paper, entitled The Sand Reckoner, was addressed to a local
monarch, and in it Archimedes claimed that he had developed a system of numbers

2 In case you’re curious, phlogiston was the hypothetical principle of fire, of which every combustible
substance was in part composed—at least until the whole theory was discredited by Antoine
Lavoisier between 1770 and 1790.
8 Chapter 1. Simple counting

that would allow him to express as large a number as the number of grains of sand in
the universe—a revolutionary idea at the time.
What Archimedes had developed was similar to what we would call exponential
notation. We’ll try to illustrate this by expressing a really large number—say, the
approximate number of seconds in the lifetime of the universe.
The calculation is simple enough. There are 60 seconds in a minute, and 60 minutes
in an hour, so the number of seconds in an hour is

60 × 60 = 3,600.

There are in turn 24 hours in a day, so the number of seconds in a day is

3,600 × 24 = 86,400;

and since there are 365 days in a (non-leap) year, the number of seconds in a year is

86,400 × 365 = 31,536,000.

Now, in exponential notation, we would say this number is roughly 3 times 10 to the
7th power—that is, a three with seven zeros after it. Here 107 refers to the product 10 ×
10 × 10 × 10 × 10 × 10 × 10 of 10 with itself seven times. In standard decimal notation,
107 = 10,000,000, a one with seven zeros after it, and thus 3 × 107 = 30,000,000 is
a three with seven zeros after it. (A better approximation, of course, would be to say
the number is roughly 3.1 × 107 , or 3.15 × 107 , but we’re going to go with the simpler
estimate 3 × 107 .)
Exponential notation is particularly convenient when it comes to multiplying large
numbers. Suppose, for example, that we have to multiply 106 × 107 . Well, 106 is just
10 × 10 × 10 × 10 × 10 × 10, and 107 is just 10 × 10 × 10 × 10 × 10 × 10 × 10, so when
we multiply them we just get the product of 10 with itself 13 times: that is,

106 × 107 = 1013 .

In other words, we simply add the exponents. So it’s easy to take products of quantities
that you’ve expressed in exponential notation.
For example, to take the next step in our problem, we have to say how old the
universe is. Now, this quantity very much depends on your model of the universe. Most
astrophysicists estimate that the universe is approximately 13.7 billion years old, with
a possible error on the order of 1%. We’ll write the age of the universe, accordingly, as

13,700,000,000 = 1.37 × 1010

years. So the number of seconds in the lifetime of the universe would be approximately

(1.37 × 1010 ) × (3 × 107 ) = 4.11 × 1017 ;

or, rounding it off, the universe is roughly 4 × 1017 seconds old.

You see how we can use this notation to express arbitrarily large numbers. For exam-
ple, computers currently can carry out on the order of 1012 operations a second (a
teraflop, as it’s known in the trade). We could ask: if such a computer were running
 1.5. It could be worse 9

from the dawn of time to the present, how many operations could it have performed?
The answer is, approximately,

1012 × (4 × 1017 ) = 4 × 1029 .

Now, for almost all of this book, we’ll be dealing with much much smaller num-
bers than these, and we’ll be doing exact calculations rather than approximations.
But occasionally we will want to express and estimate larger numbers like these. (The
last number above—the number of operations a computer running for the lifetime of
the universe could perform—will actually arise later on in this book: we’ll encounter
mathematical processes that require more than this number of operations to carry
out.) But even if there aren’t enough seconds since the dawn of time to carry out such
a calculation, it’s nice to know that we have a notation that can accommodate it.
Exercise 1.4.1. We computed the approximate age of the universe—roughly 4 × 1017
seconds old—back in 2002, so our calculation is several years old. Update our work to
compute the approximate age of the universe in seconds as of today.
Exercise 1.4.2. The Library of Alexandria is estimated to have held as many as 400,000
books (really papyrus scrolls), while the US Library of Congress currently holds about
2.8 × 106 books. How many volumes is this in total?

1.5 IT COULD BE WORSE

Look: this is a math book. We’re trying to pretend it isn’t, but it is. That means that it’ll
have jargon—we’ll try to keep it to a minimum, but we can’t altogether avoid using
technical terms. That means that you’ll encounter the odd mathematical formula here
and there. That means it’ll have long discussions aimed at solving artificially posed
problems, subject to seemingly arbitrary hypotheses. Mathematics texts have a pretty
bad reputation, and we’re sorry to say it’s largely deserved.
Just remember: it could be worse. You could, for example, be reading a book on
Kant. Now, Immanuel Kant is a towering figure in Western philosophy, a pioneering
genius who shaped much of modern thought. “The foremost thinker of the Enlight-
enment and one of the greatest philosophers of all time,” the Encyclopedia Britannica
calls him. But just read a sentence of his writing:

If we wish to discern whether anything is beautiful or not, we do not refer

the representation of it to the Object by means of understanding with a view
to cognition, but by means of the imagination (acting perhaps in conjunction
with understanding) we refer the representation to the Subject and its feeling
of pleasure or displeasure.

What’s more, this is not a nugget unearthed from deep within one of Kant’s books. It
is, in fact, the first sentence of the first Part of the first Moment of the first Book of the
first Section of Part I of Kant’s The Critique of Judgement.
Now, we’re not trying to be anti-intellectual here, or to take cheap shots at other dis-
ciplines. Just the opposite, in fact: what we’re trying to say is that any body of thought,
once it progresses past the level of bumper-sticker catchphrases, requires a language
and a set of conventions of its own. These provide the precision and universality that
are essential if people are to communicate and develop the ideas further, and shape
10 Chapter 1. Simple counting

them into a coherent whole. But they also can have the unfortunate effect of making
much of the material inaccessible to a casual reader. Mathematics suffers from this—
as do most serious academic disciplines.
The point, in other words, is not that the passage from Kant we just quoted is babble;
it’s not. (Lord knows we could have dug up enough specimens of academic writing
that are, if that was our intention.) In fact, it’s the beginning of a serious and extremely
influential attempt to establish a philosophical theory of aesthetics. As such, it may be
difficult to understand without some mental effort. It’s important to bear in mind that
the apparent obscurity of the language is a reflection of this difficulty, not necessarily
the cause of it.
So, the next time you’re reading this book and you encounter a term that turns out
to have been defined—contrary to its apparent meaning—some 30 pages earlier, or a
formula that seems to come out of nowhere and that you’re apparently expected to
find self-explanatory, just remember: it could be worse.
 2 The multiplication principle

2.1 CHOICES

Let’s suppose you climb out of bed one morning, still somewhat groggy from the night
before. You grope your way to your closet, where you discover that your cache of clean
clothes has been reduced to four shirts and three pairs of pants. It’s far too early to
exercise any aesthetic judgment whatsoever: any shirt will go with any pants; you only
need something that will get you as far as the cafe around the corner and that blessed,
life-giving cup of coffee. The question is:
How many different outfits can you make out of your four shirts and three
pairs of pants?
Admittedly the narrative took a sharp turn toward the bizarre with that last sen-
tence. Why on earth would you or anyone care how many outfits you can make? Well,
bear with us while we try to answer it anyway.
Actually, if you thought about the question at all, you probably have already fig-
ured out the answer: each of the four shirts is part of exactly three outfits, depending
on which pants you choose to go with it, so the total number of possible outfits is
3 × 4 = 12. (Or, if you like to get dressed from the bottom up, each of the three pairs
of pants is part of exactly four outfits; either way the answer is 3 × 4.) If we’re feeling
really fussy, we could make a table: say the four shirts are a polo shirt, a button-down, a
tank top, and a T-shirt extolling the virtues of your favorite athletic wear, and the pants
consist of a pair of jeans, some cargo pants, and a pair of shorts. Then we can arrange
the outfits in a rectangle:

polo shirt button-down tank top T-shirt

& jeans & jeans & jeans & jeans
polo shirt button-down tank top T-shirt
& cargo pants & cargo pants & cargo pants & cargo pants
polo shirt button-down tank top T-shirt
& shorts & shorts & shorts & shorts

Now, you know we’re not going to stop here. Suppose next that, in addition to pick-
ing a shirt and a pair of pants, you also have to choose between two pairs of shoes. Now
how many outfits are there?
Well, the idea is pretty much the same: for each of the possible shirt/pants
combinations, there are two choices for the shoes, so the total number of outfits is
11
 12 Chapter 2. The multiplication principle

4 × 3 × 2 = 12 × 2 = 24. And if in addition we had a choice of five hats, the total

number of possible outfits would be 4 × 3 × 2 × 5 = 120—you get the idea.
Now it’s midday and you head over to the House of Pizza to order a pizza for lunch.
You feel like having one meat topping and one vegetable topping on your pizza; the
House of Pizza offers you seven meat toppings and four vegetable toppings. How many
different pizzas do you have to choose among?
“That’s the same problem with different numbers!” you might say, and you’d be
right: to each of the seven meat toppings you could add any one of the four vegetable
toppings, so the total number of different pizzas you could order would be 7 × 4, or 28.
Evening draws on, and your roommates ask you to queue up a triple feature to
watch. They request one action film, one romantic comedy, and one comedy special.
The film streaming service you subscribe to has 674 action films (most of which were
direct-to-video), 913 romantic comedies (ditto), and 84 comedy specials. How many
triple features can you watch?
“That’s the same problem again!” you might be thinking: the answer’s just the num-
ber 674 of action movies times the number 913 of romantic comedies times the num-
ber 84 of comedy specials, or

674 × 913 × 84 = 51,690,408.

Trust us—we are going somewhere with this. But you’re right, it’s time to state the
general rule that we’re working toward, which is called the multiplication principle:

The number of ways of making a sequence of independent choices is the product

of the number of choices at each step.

Here “independent” means that how you make the first choice doesn’t affect the
number of choices you have for the second, and so on. In the first case above, for
example—getting dressed in the morning—it corresponds to having no fashion sense
whatsoever.
The multiplication principle is easy to understand and apply, but awkward to state
in reasonably coherent English, which is why we went through three examples before
announcing it. In fact, you may find the examples more instructive than the principle
itself; if the boxed statement seems obscure to you, just remember: “4 shirts and 3
pants equals 12 outfits.”
Exercise 2.1.1. You have seven colors of nail polish and 10 fingers. You’re not coordi-
nated enough to use more than one color on each nail, but aren’t fussed about using
different colors on different nails. How many ways can you paint the nails on your
hands if you are not worried about whether they match?

2.2 COUNTING WORDS

An old-style license plate has on it a sequence of three numbers followed by three

letters. How many different old-style license plates can there be?
This is easy enough to answer: we have 10 choices for each of the numbers, and 26
choices for each of the letters; and since none of these choices is constrained in any
way, the total number of possible license plates is
2.2. Counting words 13

10 × 10 × 10 × 26 × 26 × 26 = 17,576,000.

A similar question is this: Suppose for the moment that by “word” we mean any
finite sequence of the 26 letters of the English alphabet—we’re not going to make
a distinction between actual words and arbitrary sequences. How many three-letter
words are there?
This is just the same as the license plate problem (or at least the second half ): we
have 26 independent choices for each of the letters, so the number of three-letter
words is 263 = 17,576. In general,

# of 1-letter words = 26,

# of 2-letter words = 262 = 676,
# of 3-letter words = 263 = 17,576,
# of 4-letter words = 264 = 456,976,
# of 5-letter words = 265 = 11,881,376,
# of 6-letter words = 266 = 308,915,776,

and so on.
Next, let’s suppose that there are 15 students in a class, and that they’ve decided to
choose a set of class officers: a president, a vice president, a secretary and a treasurer.
How many possible slates are there? That is, how many ways are there of choosing the
four officers?
Actually, there are two versions of this question, depending on whether or not a
single student is allowed to hold more than one of the positions. If we assume first
that there’s no restriction of how many positions one person can hold, the problem
is identical to the ones we’ve just been looking at: we have choices each for the four
offices, and they are all independent, so that the total number of possible choices is

15 × 15 × 15 × 15 = 50,625.

Now suppose on the other hand that we impose the rule that no person can hold
more than one office. How many ways are there of choosing officers now?
Well, this can also be computed by the multiplication principle. We start by choos-
ing the president; we have clearly 15 choices there. Next, we choose the vice president.
Now our choice is restricted by the fact that our newly selected president is no longer
eligible, so that we have to choose among the 14 remaining students. After that we
choose a secretary, who could be anyone in the class except the two officers already
chosen, so we have 13 choices here; finally we choose a treasurer from among the 12
students in the class other than the president, vice president, and secretary. Altogether,
the number of choices is

15 × 14 × 13 × 12 = 32,760.

Note one point here: in this example, the actual choice of, say, the vice-president
does depend on who we chose for president; the choice of a secretary does depend on
who we selected for president and vice president, and so on. But the number of choices
doesn’t depend on our prior selections, so the multiplication principle still applies.
In a similar vein, we could modify the question we asked a moment ago about the
number of three-letter words, and ask: how many three-letter words have no repeated
14 Chapter 2. The multiplication principle

letters? The solution is completely analogous to the class-officer problem: we have 26

choices for the first letter, 25 for the second, and 24 for the third, so that we have a
total of
26 × 25 × 24 = 15,600
such words.
In general, we can calculate

# of 1-letter words = 26,

# of 2-letter words without repeated letters = 26 · 25 = 650,
# of 3-letter words without repeated letters = 26 · 25 · 24 = 15,600,
# of 4-letter words without repeated letters = 26 · 25 · 24 · 23 = 358,800,
# of 5-letter words without repeated letters = 26 · 25 · 24 · 23 · 22
= 7,893,600,
# of 6-letter words without repeated letters = 26 · 25 · 24 · 23 · 22 · 21
= 165,765,600,

and so on. Note that here we are using a simple dot · in place of the times symbol ×. In
general, when we have an expression with a lot of products, we’ll use this simpler nota-
tion to avoid clutter. Sometimes we’ll omit the product sign altogether; for example, we
write 2n for 2 × n.
Now, here’s an interesting (if somewhat tangential) question. Let’s compare the
numbers of words of each length to the number of words with no repeated letters.
What percentage of all words have repetitions, and what percentage doesn’t? Of course,
as the length of the word increases, we’d expect a higher proportion of all words to have
repeated letters—relatively few words of two or three letters have repetitions, while
necessarily every word of 27 or more letters does. We could ask, then: when does the
fraction of words without repeated letters dip below one-half? In other words, for what
lengths do the words with repeated letters outnumber those without?
Before we tabulate the data and give the answer, you might want to take a few
minutes and think about the question. What would your guess be?

length number of words without repeats % without repeats

1 26 26 100.00
2 676 650 96.15
3 17,576 15,600 88.76
4 456,976 358,800 78.52
5 11,881,376 7,893,600 66.44
6 308,915,776 165,765,600 53.66
7 8,031,810,176 3,315,312,000 41.28
8 208,827,064,576 62,990,928,000 30.16
9 5,429,503,678,976 1,133,836,704,000 20.88

Now that’s bound to be surprising: among six-letter words, those with repeated
letters represent nearly half, and among seven-letter words they already substantially
outnumber the words without repeats. In general, the percentage of words without
 2.3. A sequences of choices 15

repeated letters drops off pretty fast: by the time we get to twelve-letter words, fewer
than 1 in 20 has no repeated letter. We’ll see another example of this phenomenon
when we talk about the birthday problem in Section 5.6.
Exercise 2.2.1. Suppose you want to select a class president, vice president,
secretary, and treasurer, under the condition that no person can hold more than
one office, but this time the plan is to choose the treasurer first, then the secre-
tary, then the vice president, and then the president. How many ways are there of
choosing the four officers now? How does this relate to the computation performed
above?
Exercise 2.2.2. The Greek alphabet has 24 letters while the Russian alphabet consists
of 33 letters.
1. Predict whether the percentage of words of length n in the Greek alphabet without
repeats will be smaller or greater than the percentage of words of length n in the
English alphabet without repeats.
2. Predict whether the percentage of words of length n in the Russian alphabet without
repeats will be smaller or greater than the percentage of words of length n in the
English alphabet without repeats.
3. Compute the number of words of length 1, 2, 3, 4, and 5 with and without repeats in
Greek and in Russian and see whether or not your predictions were correct.

2.3 A SEQUENCES OF CHOICES

There are two special cases of the multiplication principle that occur so commonly in
counting problems that they’re worth mentioning on their own, and we’ll do that here.
Neither will be new to us; we’ve already encountered examples of each.
Both involve a sequence of selections from a single pool of objects. If there are
no restrictions at all on the choices, the application of the multiplication principle is
particularly simple: each choice in the sequence is a choice among all the objects in
the collection. If we’re counting three-letter words in an alphabet of 26 characters, for
example—where by “word” we again mean an arbitrary sequence of letters—there are
263 ; if we’re counting four-letter words in an alphabet of 22 characters, there are 224 ;
and so on. In general, we have the following rule:

The number of sequences of k objects chosen from a collection of n objects is nk .

The second special case involves the same problem, but with a commonly applied
restriction: we’re again looking at sequences of objects chosen from a common pool
of objects, but this time we’re not allowed to choose the same object twice. Thus, the
first choice is among all the objects in the pool; the second choice is among all but
one, the third among all but two, and so on; if we’re looking at a sequence of k choices,
the last choice will be among all but the k − 1 already chosen. Thus, as we saw, the
number of three-letter words without repeated letters in an alphabet of 26 characters
is 26 · 25 · 24; the number of four-letter words without repeated letters in an alphabet of
 16 Chapter 2. The multiplication principle

22 characters is 22·21·20·19; and so on. In general, if the number of objects in our pool
is n, the first choice will be among all n, the second among n − 1, and so on. If we’re
making a total of k choices, the last choice will exclude the k −1 already chosen; that is,
it’ll be a choice among the n − (k − 1) = n − k + 1 objects remaining. The total number
of such sequences is thus the product of the numbers from n down to n − k + 1. We
write this as
n · (n − 1) · (n − 2) · · · · · (n − k + 1),
where the dots in the middle indicate that you’re supposed to keep going multiplying
all the whole numbers in the series starting with n, n − 1 and n − 2, until you get down
to n − k + 1. Time for a box:

The number of sequences of k objects chosen without repetition from a collection

of n objects is n · (n − 1) · (n − 2) · · · · · (n − k + 1).

Exercise 2.3.1. In the state lottery, the winning number is chosen by picking six ping-
pong balls from a bin containing balls labeled “1” through “36” to arrive at a sequence
of six numbers between 1 and 36. Ping-pong balls are not replaced after they’re chosen;
that is, no number can appear twice in the sequence. How many possible outcomes are
there?
Note that in this last exercise, the order in which the ping-pong balls are chosen is
relevant: if the winning sequence is “17-32-5-19-12-27” and you picked “32-17-5-19-
12-27,” you don’t get to go to work the next day and tell your boss what you really think
of her.
Exercise 2.3.2. The Hebrew alphabet has 22 letters. How many five-letter words are
possible in Hebrew? (Again, by “word” we mean just an arbitrary sequence of five
characters from the Hebrew alphabet.) What fraction of these have no repeated letters?

2.4 FACTORIALS

The two formulas we described in the last section were both special cases of the multi-
plication principle. There is in turn a special case of the second formula that crops up
fairly often and that’s worth talking about now. We’ll start, as usual, with an example.
Problem 2.4.1. Suppose that we have a first-grade class of 15 students, and we want
to line them up to go out to recess. How many ways of lining them up are there—that
is, in how many different orders can they be lined up?

Solution. Well, think of it this way: we have 15 choices of who’ll be first in line. Once
we’ve chosen the line leader, we have 14 choices for who’s going to be second, 13
choices for the third, and so on. In fact, all we’re doing here is choosing a sequence
of 15 children from among the 15 children in the class, without repetition; whether we
invoke the formula in the last section or do it directly, the answer is

15 · 14 · 13 · 12 · 11 · 10 · 9 · 8 · 7 · 6 · 5 · 4 · 3 · 2 · 1 = 1,307,674,368,000,

or about 1.3 × 1012 —more than a trillion orderings.

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THE GOLDEN STAIRCASE 325 A CHRISTMAS CAROL '

WHAT means this glory round our feet,' The Magi mused, ' more
bright than morn ? ' And voices chanted clear and sweet, 1 To-day
the Prince of Peace is born.' ' What means that star,' the shepherds
said, ' That brightens through the rocky glen ? ' And angels,
answering overhead, Sang ' Peace on earth, good-will to men.' 'Tis
eighteen hundred years and more Since those sweet oracles were
dumb ; We wait for Him, like them of yore ; Alas ! He seems so slow
to come. But it was said in words of gold, No time or sorrow e'er
shall dim, That little children might be bold, In perfect trust to come
to Him. All round about our feet shall shine A light like that the wise
men saw, If we our willing hearts incline To that sweet Life which is
the Law. So shall we learn to understand The simple faith of
shepherds then, And, kindly clasping hand in hand, Sing, ' Peace on
earth, good- will to men.' For they who to their childhood cling, And
keep their natures fresh as morn, Once more shall hear the angels
sing, ' To-day the Prince of Peace is born.' JAMES RUSSELL LOWELL.
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326 THE GOLDEN STAIRCASE A CHRISTMAS CAROL IN the

bleak mid-winter Frosty wind made moan, Earth stood hard as iron,
Water like a stone ; Snow had fallen, snow on snow, Snow on snow,
In the bleak mid-winter Long ago. Our God, Heaven cannot hold
Him, Nor earth sustain ; Heaven and earth shall flee away When He
comes to reign : In the bleak mid- winter A stable-place sufficed The
Lord God Almighty, Jesus Christ. Enough for Him whom cherubim
Worship night and day, A breastf ul of milk And a mangerf ul of hay ;
Enough for Him whom angels Fall down before, The ox and ass and
camel Which adore. Angels and archangels May have gathered
there, Cherubim and Seraphim Throng'd the air ; But only His
mother In her maiden bliss Worshipped the Beloved With a kiss.
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THE GOLDEN STAIRCASE 327 What can I give Him, Poor as

I am ? If I were a shepherd I would bring a lamb, If I were a wise
man I would do my part, — Yet what I can I give Him, Give my
heart. CHRISTINA G. ROSSETTI. THE THREE KINGS OF COLOGNE
FROM out Cologne there came three kings To worship Jesus Christ
their King. To Him they sought, fine herbs they brought, And many a
beauteous golden thing ; They brought their gifts to Bethlehem
town, And in that manger set them down. Then spake the first king,
and he said : 1 O Child, most heavenly, bright and fair 1 I bring this
crown to Bethlehem town For Thee, and only Thee, to wear ; So
give a heavenly crown to me When I shall come at last to Thee ! '
The second then : ' I bring Thee here This royal robe, O Child ! ' he
cried ; • Of silk 'tis spun, and such an one There is not in the world
beside ; So in the day of Doom requite Me with a heavenly robe of
white ! ' The third king gave his gift and quoth : ' Spikenard and
myrrh to Thee I bring, And with these twain would I most fain
Anoint the body of my King ; So may their incense sometime rise To
plead for me in yonder skies ! '
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328 THE GOLDEN STAIRCASE Thus spake the three kings

of Cologne, That gave their gifts and went their way ; And now
kneel I in prayer hard by The cradle of the Child to-day ; Nor crown,
nor robe, nor spice I bring As offering unto Christ, my King. Yet have
I brought a gift this Child May not despise, however small ; For here
I lay my heart to-day, And it is full of love to all. Take, then, this poor
but loyal thing, My only tribute, Christ, my King. EUGENE FIELD. A
CHRISTMAS HYMN ONCE in royal David's city Stood a lowly cattle-
shed Where a mother laid her Baby, In a manger for His bed. Mary
was that mother mild, Jesus Christ her little Child. He came down to
earth from heaven, Who is God and Lord of all, And His shelter was
a stable, And His cradle was a stall. With the poor, and mean, and
lowly Lived on earth our Saviour holy. And through all His wondrous
childhood, He would honour and obey, Love and watch the lowly
mother In whose gentle arms He lay. Christian children, all must be
Mild, obedient, good as He.
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THE GOLDEN STAIRCASE 329 For He is our childhood's

Pattern, Day by day like us He grew ; He was little, weak, and
helpless, Tears and smiles like us He knew ; And He feeleth for our
sadness, And He shareth in our gladness. And our eyes at last shall
see Him, Through His own redeeming love, For that Child so dear
and gentle Is our Lord in Heaven above ; And He leads His children
on To the place where He is gone. Not in that poor lowly stable,
With the oxen standing by, We shall see Him ; but in Heaven, Set at
God's right hand on high ; When like stars His children crowned, All
in white shall wait around. 0. FRANCES ALEXANDER. THE CHILD OF
BETHLEHEM O LITTLE town of Bethlehem, How still we see thee lie !
Above thy deep and dreamless sleep The silent stars go by ; Yet in
thy dark streets shineth The everlasting light ; The hopes and fears
of all the years Are met in thee to-night ! For Christ is born of Mary ;
And gathered all above, While mortals sleep, the angels keep Their
watch of wondering love.
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330 THE GOLDEN STAIRCASE O morning stars ! together

Proclaim the holy birth, And praises sing to God the King, And peace
to men on earth ! How silently, how silently, The wondrous gift is
given 1 So God imparts to human hearts The blessings of His
heaven. No ear may hear His coming ; But in this world of sin,
Where meek souls will receive Him, still The dear Christ enters in. O
holy Child of Bethlehem ! Descend to us, we pray ; Cast out our sin
and enter in — Be born in us to-day ! We hear the Christmas angels
The great glad tidings tell ; Oh, come to us, abide with us, Our Lord
Emmanuel ! PHILLIPS BROOKS. CHRISTMAS DAY A BABY is a
harmless thing, And wins our heart with one accord, And Flower of
Babies was their King, Jesus Christ our Lord : Lily of lilies He Upon
His Mother's knee ; Rose of roses, soon to be Crowned with thorns
on leafless tree. A lamb is innocent and mild, And merry on the soft
green sod ; And Jesus Christ, the Undefiled, Is the Lamb of God :
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THE GOLDEN STAIRCASE 331 Only spotless He Upon His

Mother's knee ; White and ruddy, soon to be Sacrificed for you and
me. Nay, lamb is not so sweet a word, Nor lily half so pure a name ;
Another name our hearts hath stirred. Kindling them to flame : *
Jesus ' certainly Is music and melody : Heart with heart in harmony
Carol we and worship we. CHRISTINA ROSSETTI. NEW PRINCE,
NEW POMP BEHOLD a silly l tender Babe, In freezing winter night,
In homely manger trembling lies ; Alas, a piteous sight 1 The inns
are full, no man will yield This little Pilgrim bed ; But forc'd He is
with silly beasts In crib to shroud His head. Despise not Him for lying
there, First what He is enquire ; An orient pearl is often found In
depth of dirty mire. Weigh not His crib, His wooden dish, Nor beasts
that by Him feed ; Weigh not His mother's poor attire, Nor Joseph's
simple weed. 1 ' Silly' here means 'innocent.'
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332 THE GOLDEN STAIRCASE This stable is a Prince's

court, The crib His chair of State ; The beasts are parcel of His
pomp, The wooden dish His plate. The persons in that poor attire
His royal liveries wear ; The Prince Himself is come from heaven,
This pomp is prized there. With joy approach, O Christian wight, Do
homage to thy King ; And highly prize His humble pomp, Which He
from heaven doth bring. ROBERT SOUTHWELL. A HYMN OF THE
NATIVITY • • t • • • GLOOMY night embraced the place Where the
noble Infant lay. The Babe look'd up and show'd His face ; In spite of
darkness, it was day. It was Thy day, Sweet ! and did rise Not from
the East, but from Thine eyes. Winter chid aloud, and sent The
angry North to wage his wars. The North forgot his fierce intent, And
left perfumes instead of scars. By those sweet eyes' persuasive
powers, Where he meant frost, he scatter'd flowers. We saw Thee in
Thy balmy nest, Young dawn of our Eternal Day ; We saw Thine
eyes break from their East And chase the trembling shades away.
We saw Thee ; and we blest the sight, We saw Thee by Thine own
sweet light.
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THE GOLDEN STAIRCASE 333 Poor world, said I, what wilt

thou do To entertain this Starry Stranger? Is this the best thou canst
bestow? A cold, and not too cleanly, manger ? Contend, the powers
of Heaven and Earth, To fit a bed for this huge birth? I saw the
curl'd drops, soft and slow, Come hov'ring o'er the place's head ;
Off'ring their whitest sheets of snow To furnish the fair Infant's bed :
Forbear, said I ; be not too bold, Your fleece is white, but 'tis too
cold. I saw the obsequious Seraphims Their rosy fleece of fire
bestow, For well they now can spare their wing, Since Heaven itself
lies here below. Well done, said I ; but are you sure Your down so
warm, will pass for pure ? No, no ! your King 's not yet to seek
Where to repose His royal head ; See, see ! how soon His new-
bloom'd cheek 'Twixt 's mother's breasts is gone to bed. Sweet
choice ! said we, no way but so Not to lie cold, yet sleep in snow.
Welcome, all wonders in one sight ! Eternity shut in a span !
Summer in Winter, Day in Night ! Heaven in Earth, and God in man !
Great little One ! whose all-embracing birth Lifts Earth to Heaven,
stoops Heaven to Earth. RICHARD CRASHAW.
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334 THE GOLDEN STAIRCASE HOW CHRISTMAS CAME

HEAVEN'S fairest star Trembled a moment in the gold-flecked blue ;
Then, earthward dropped, Was in an empty cradle lost to view, Till
angel came, And softly parting back the curtains, smiled, While hosts
proclaimed The birth of Bethlehem's King in new-born child. CALLIE
L. BONNEY. TO HIS SAVIOUR, A CHILD; A PRESENT, BY A CHILD Go,
prettie child, and beare this Flower Unto thy little Saviour ; And tell
Him, by that Bud now blown, He is the Rose of Sharon known :
When thou hast said so, stick it there Upon His Bibb, or Stomacher :
And tell Him (for good handsell too) That thou hast brought a
Whistle new, Made of a clean strait oaten reed, To charme His cries
(at time of need :) Tell Him, for Corall, thou hast none ; But if thou
hadst, He sho'd have one ; But poore thou art, and knowne to be
Even as monilesse, as He. Lastly, if thou canst win a kisse From
those mellifluous lips of His ; Then never take a second on, To spoile
the first impression. ROBERT HERRICK.
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THE GOLDEN STAIRCASE 335 THE STAR SONG : A CAROLL

TO THE KING TELL us, thou cleere and heavenly Tongue, Where is
the Babe but lately sprung ? Lies He the Lillie-banks among ? Or say,
if this new Birth of ours Sleeps, laid within some Ark of Flowers,
Spangled with deaw-light ; thou canst cleere All doubts, and
manifest the where. Declare to us, bright Star, if we shall seek Him
in the Morning's blushing cheek, Or search the beds of Spices
through, To find Him out ? STAB No, this ye need not do ; But only
come, and see Him rest A Princely Babe in 's Mother's Brest.
CHORUS Come then, come then, and let us bring Unto our prettie
Twelfth-Tide King, Each one his severall offering ; And when night
comes, we 11 give Him wassailing ; And that His treble Honours may
be seen, We 11 chuse Him King, and make His Mother Queen.
ROBERT HERRICK. A CAROL FOR CHRISTMAS EVE LISTEN,
lordlings, unto me, a tale I will you tell, Which, as on this night of
glee, in David's town befell.
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336 THE GOLDEN STAIRCASE Joseph came from Nazareth,

with Mary, that sweet maid: Weary were they, nigh to death ; and
for a lodging pray'd. Sing high, sing low, sing to and fro, Go tell it
out with speed, Cry out and shout all round about, That Christ is
born indeed. In the inn they found no room; a scanty bed they
made: Soon a Babe from Heaven high was in the manger laid. Forth
He came, maid Mary's Son : He came to save us all. In the stable, ox
and ass before their Maker fall. Sing high, sing low, sing to and fro,
Go tell it out with speed, Cry out and shout all round about, That
Christ is born indeed. Shepherds lay afield that night, to keep the
silly sheep, Hosts of Angels in their sight came down from heaven's
high steep. Tidings ! Tidings ! unto you : to you a Child is born,
Purer than the drops of dew, and brighter than the morn. Sing high,
sing low, sing to and fro, Go tell it out with speed, Cry out and shout
all round about, That Christ is born indeed. Onward then the Angels
sped, the shepherds onward went, God was in His manger-bed, in
worship low they bent. In the morning, see ye mind, my masters
one and all, At the Altar Him to find who lay within the stall. Sing
high, sing low, sing to and fro, Go tell it out with speed, Cry out and
shout all round about, That Christ is born indeed. UNKNOWN.
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THE GOLDEN STAIRCASE 337 THE NEW-YEERE'S GIFT LET

others looke for Pearle and Gold, Tissues, or Tabbies manifold : One
onely lock of that sweet Hay Whereon the blessed Babie lay, Or one
poore Swadling-clout, shall be The richest New-Yeere's Gift to me.
ROBERT HERRICK. A LITTLE CHILD'S HYMN THOU that once, on
mother's knee, Wert a little one like me, When I wake or go to bed,
Lay Thy hands about my head ; Let me feel Thee very near, Jesus
Christ, our Saviour dear. Be beside me in the light, Close by me
through all the night ; Make me gentle, kind, and true, Do what
mother bids me do ; Help and cheer me when I fret, And forgive
when I forget. Once wert Thou in cradle laid, Baby bright in manger-
shade, With the oxen and the cows, And the lambs outside the
house : Now Thou art above the sky ; Canst Thou hear a baby cry ?
Thou art nearer when we pray, Since Thou art so far away ; Thou
my little hymn wilt hear, Jesus Christ, our Saviour dear, Thou that
once, on mother's knee, Wert a little one like me. FRANCIS TURNER
PALGRAVE. Y
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338 THE GOLDEN STAIRCASE MORNING HYMN Now the

sun is in the skies, From my bed again I rise ; Christ, Thou never-
setting Sun, Shine on me, Thy little one. Watch me through the
coming day, Guard me in my work and play ; Christ my Master,
Christ the Child, Make me like Thee, Jesu mild. Christ, Almighty King
above, Thee I pray for all I love ; Christ, who lovest more than I,
Help them from Thy throne on high. Christ, of Mary born for me, To
Thy name I bow the knee ; Saviour, bring us, by Thy grace, To Thy
happy dwelling-place. B. P. LITTLEDALR THE GOOD SHEPHERD
KIND Shepherd, see, Thy little lamb Comes very tired to Thee ; 0
fold me in Thy loving arms, And smile on me. 1 Ve wander'd from
Thy fold to-day, And could not hear Thee call ; And O ! I was not
happy then, Nor glad at all.
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THE GOLDEN STAIRCASE 339 I want, dear Saviour, to be

good, Aud follow close to Thee, Through flowery meads and
pastures green And happy be. Thou kind, good Shepherd, in Thy
fold I evermore would keep, In morning's light or evening's shade,
And while I sleep. But now, dear Jesus, let me lay My head upon
Thy breast ; I am too tired to tell Thee more, Thou know'st the rest.
H. P. HAWKINS. EVENING HYMN Now the day is over, Night is
drawing nigh, Shadows of the evening Steal across the sky. Now the
darkness gathers, Stars begin to peep, Birds, and beasts, and
flowers Soon will be asleep. Jesu, give the weary Calm and sweet
repose ; With Thy tenderest blessing May our eyelids close. Grant to
little children Visions bright of Thee; Guard the sailors tossing On the
deep blue sea.
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340 THE GOLDEN STAIRCASE Comfort every sufferer

Watching late in pain ; Those who plan some evil, From their sin
restrain. Through the long night-watches May Thine angels spread
Their white wings above me, Watching round my bed. When the
morning wakens, Then may I arise Pure, and fresh, and sinless In
Thy holy eyes. 8. BARING-GOULD, THE TENDER SHEPHERD JESUS,
tender Shepherd, hear me : Bless Thy little lamb to-night ; Through
the darkness be Thou near me, Keep me safe till morning light.
Through this day Thy hand hath led me, And I thank Thee for Thy
care ; Thou hast warmed me, clothed, and fed me, Listen to my
evening prayer. Let my sins be all forgiven, Bless the friends I love
so well ; Take me, when I die, to heaven, Happy, there, with Thee to
dwell. MARY L. DUNCAN.
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THE GOLDEN STAIRCASE 341 A CHILD'S PRAYER GOD,

make my life a little light Within the world to glow ; A little flame
that burneth bright, Wherever I may go. God, make my life a little
flower That giveth joy to all, Content to bloom in native bower,
Although the place be small. God, make my life a little song That
comf orteth the sad ; That helpeth others to be strong, And makes
the singer glad. God, make my life a little staff Whereon the weak
may rest, That so what health and strength I have May serve my
neighbours best. God, make my life a little hymn Of tenderness and
praise ; Of faith — that never waxeth dim, In all His wondrous ways.
MATILDA B. EDWARDS. JESUS BIDS US SHINE JESUS bids us shine
With a pure clear light, Like a little candle Burning in the night ; In
the world is darkness, So we must shine — You in your small corner,
And I in mine.
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342 THE GOLDEN STAIRCASE Jesus bids us shine First of

all for Him : Well He sees and knows it, If our light grows dim ; He
looks down from heaven To see us shine — You in your small corner,
And I in mine. Jesus bids us shine, Then, for all around : Many kinds
of darkness In the world are found ; Sin and want and sorrow ; So
we must shine — You in your small corner, And I in mine. EMILY H.
MILLER. ALL THINGS BRIGHT AND BEAUTIFUL ALL things bright
and beautiful, All creatures great and small, All things wise and
wonderful, The Lord God made them all. Each little flower that
opens, Each little bird that sings, He made their glowing colours, He
made their tiny wings. The rich man in his castle, The poor man at
his gate, God made them, high or lowly, And order 'd their estate.
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THE GOLDEN STAIRCASE 343 The purple-headed

mountain, The river running by, The sunset and the morning, That
brightens up the sky. The cold wind in the winter, The pleasant
summer sun, The ripe fruits in the garden, He made them every
one. The tall trees in the greenwood, The meadows where we play,
The rushes by the water, We gather every day ;— He gave us eyes
to see them, And lips that we might tell How great is God Almighty,
Who has made all things well. C. FRANCES ALEXANDER. GOD, WHO
HATH MADE THE DAISIES GOD, who hath made the daisies And
ev'ry lovely thing, He will accept our praises, And hearken while we
sing. He says though we are simple, Though ignorant we be, 1
Suffer the little children, And let them come to Me.' Though we are
young and simple, In praise we may be bold ; The children in the
temple He heard in days of old.
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344 THE GOLDEN STAIRCASE And if our hearts are humble,

He says to you and me, ' Suffer the little children, And let them
come to Me.' He sees the bird that wingeth Its way o'er earth and
sky ; He hears the lark that singeth Up in the heaven so high ; But
sees the heart's low breathings, And says (well pleased to see), 1
Suffer the little children, And let them come to Me.' Therefore we
will come near Him, And solemnly we '11 sing ; No cause to shrink
or fear Him, We '11 make our voices ring ; For in our temple
speaking, He says to you and me, ' Suffer the little children, And let
them come to Me.' E. P. HOOD. PSALM XXIII THE God of love my
Shepherd is, And He that doth me feed, While He is mine, and I am
His, What can I want or need ? He leads me to the tender grass,
Where I both feed and rest ; Then to the streams that gently pass :
In both I have the best.
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THE GOLDEN STAIRCASE 345 Or if I stray, He doth convert,

And bring my mind in frame : And all this not for my desert, But for
His holy name. Yea, in Death's shady black abode Well may I walk,
not fear ; For Thou art with me, and Thy rod To guide, Thy staff to
bear. Nay, Thou dost make me sit and dine Ev'n in my enemies' sight
; My head with oil, my cup with wine Runs over day and night.
Surely Thy sweet and wondrous love Shall measure all my days ;
And as it never shall remove, So neither shall my praise. GEORGE
HERBERT. EARLY PIETY BY cool Siloam's shady rill How sweet the lily
grows ! How sweet the breath beneath the hill Of Sharon's dewy
rose ! Lo ! such the child whose early feet The paths of peace have
trod ; Whose secret heart, with influence sweet, Is upward drawn to
God ! By cool Siloam's shady rill The lily must decay ; The rose that
blooms beneath the hill Must shortly fade away.
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346 THE GOLDEN STAIRCASE And soon, too soon, the

wintry hour Of man's maturer age Will shake the soul with sorrow's
power, And stormy passion's rage ! O Thou, whose infant feet were
found Within Thy Father's shrine ! Whose years, with changeless
virtue crowu'd, Were all alike Divine ; Dependent on Thy bounteous
breath, We seek Thy grace alone, In childhood, manhood, age and
death, To keep us still Thine own ! BISHOP HEBER. EX ORE
INFANTIUM LITTLE JESUS, wast Thou shy Once, and just so small as
I ? And what did it feel like to be Out of Heaven and just like me ?
Didst Thou sometimes think of there, And ask where all the angels
were ? I should think that I would cry For my house all made of sky
; I would look about the air, And wonder where my angels were ;
And at waking 'twould distress me — Not an angel there to dress me
! Hadst Thou ever any toys, Like us little girls and boys ? And didst
Thou play in Heaven with all The angels that were not too tall, With
stars for marbles ? Did the things Play ' can you see me?' through
their wings?
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