2017 F-CH2-1

1. What is the total number of recognizably distinct patterns when we roll

4 dice? (A pattern is defined by which faces show up not in which order)
2. Five people, among whom X and Y, are scheduled to speak at a conference.
In how many ways can X speak immediately before Y? In how many ways
can the speakers be arranged without Y speaking before X?

3. How many ways are there to distribute among a class of 10 students three
copies of one book, 2 copies of a second book and one copy of a third
book, in such a way that a) no student receives more than one book b)
no students receives more than one copy of any book? [That is for a) a
student receives at most one book (A, B, or C) and only one copy (that
is, no student will receive BB or AB, e.g.) and for b) a student may also
receive two or three books but not of the same title, say ABC, but not
AA]
4. In how many ways can we place in a bookcase four different history books,
four different philosophy books, 3 different books of physics and 3 different
books of geometry so that books of the same subject are not separated?
5. If we use an alphabet with 8 consonants and 4 vowels, how many strings
of letters made of (exactly) 4 different consonants and 2 different vowels
can be obtained?
6. How many ways can we split {1, 2, ..., 10} into 6 subsets of size 1, 2 subsets
of size 2 [not distinguishable apart from their sizes]?
7. How many strings of length 10 can be formed from the set {a, b, c, d, e} ∪
{0, 1, . . . , 9} that have exactly 3 letters, exactly 3 0’s, and 4 non-zero
numbers?

8. What is the probability that a leap year, selected at random, will contain
fifty-three Sundays ?
9. What is the probability that on casting two dice neither 1 nor 3 appears?
10. A deck consists of 52 cards, with 13 values (A, 2, . . . , 10, J, Q, K) each
having 4 suits ♦, ♥, ♠,♣. J, Q, K are also called face cards. From a deck
of 52 cards one card is drawn at random. What is the probability that
this card is a heart (♥) or face card of an arbitrary suit?
0

11. k numbered balls can each occupy each of n labelled cells. Cells can
accommodate any number of balls. Assume all arrangements to be equi-
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probable. (a) Find the probability that there will be one ball in each of
k definite cells [that is, in k fixed and specified cells. Put it differently,
if k = 3, we want to find the probability that the three cells numbered,
say, 2, 7, 9 contain one particle.] (b) Find the probability that there will
be one ball in each of k arbitrary cells (you do not know which are these
k cells).

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12. k identistinguishable balls can each occupy each of n numbered cells, which
can accommodate any number of balls. Assuming all allocations of the
balls to be equally-probable, find the probability that: (a) there will be
one ball in each of k definite cells (b) there will be one ball in each of k
arbitrary cells.

13. k identical balls can each occupy each of n cells, and a cell can contain at
most one ball. Assuming all allocations of the balls to be equally-probable,
find the probability that: (a) there will be one ball in each of k definite
cells (b) there will be one ball in each of k arbitrary cells.
14. If n numbered balls are placed in n cells at random (with same probability
1/n), what is the probability that exactly one cell is empty?
15. A class composed of 16 men and 16 women is randomly divided into two
groups of equal size. What is the probability that each group has 8 men
and 8 women?

16. A gardener wishes to plant three pine trees, seven oak trees, and four
cedars in a row. Assuming that he plants them in random order with each
arrangement being equally probable. What is the probability that no two
cedars are next to each other?
17. Find the probability that if we sample with replacement 4 letters of a
26-letter alphabet, all letters are the same

18. Find the probability that if we sample with replacement 4 letters of a

26-letter alphabet, exactly two letters of the four are different letters
19. Find the number of total strings of length 4 from a 26-letter alphabet

20. In how many ways 4 red balls, 6 white balls, 7 blue balls can be placed in
5 labelled boxes, when one or more may be left empty?
21. Six people enter an empty elevator at the ground floor of a building with
12 floors. Assuming all arrangements of leaving the elevators to be equally
probable, what is the probability that no two passengers exit at the same
floor?

22. We generate with a compute k = 4 random number (between 0 and 9,

with replacement) what is the probability that no two are equal?
0

23. Consider a group of n people. Assuming all days of the year are equiprob-
able and there are 365 days in a year, compute the probability that a) two
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specific people, Nour and Joy, share their birthday on March 1st; b) Only
Nour and Joy were born on March 15; c) Only Nour and Joy were born
on the same day and this is March 15; d) Only Nour and Joy have the
same birthday; e) Only two people among the n share the same birthday

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24. Nathalie enters a room where there are 10 people. What is the probability
that none of these people shares her birthday, assuming all days of the year
are equiprobable and there are 365 days in a year?
25. Seven distinguishable balls are placed randomly (=all arrangements are
equi-probable) into four labelled boxes. What is the probability that no
box is left empty?
26. Five numbers are randomly selected one after another without replacement
from an urn with digits {0, . . . , 9}. (does it matter that they are sampled
one after the other or all together?) What is the probability that the
sample contains 1?

27. Four numbers are randomly selected one after another with replacement
from an urn with digits {0, . . . , 9}. What is the probability that 1 appears
once.
28. A deck consists 52 cards, with 13 values (A, 2, . . . , 10, J, Q, K) each having
4 suits ♦, ♥, ♠,♣. What is the probability of being dealt a pair at poker
(exactly a pair, that is two cards with the same value, and three others
of values different from the value of the pair and from each other: e.g.
2♥ 2♠ 3♥K♠Q♦)?
29. An urn contains 15 red and 15 blue balls. The red balls are numbered
from 1 to 15, and so are the blue. Ten balls are drawn all at once from the
urn. What is the probability that (exactly) two numbers are repeated?
30. From a regular 52-card deck-described above- cards are drawn successfully
until a K (King) appears. What’s the probability that a K will appear at
the n-th draw? What is the probability that it will appear after the n-th
draw?

31. A box contains 4 white balls, 4 blue balls, 4 red balls and 4 yellow balls.
A sample of eight balls are drawn from the box without replacement. a)
Find the probability that the sample contains two balls of each color b)
Find the probability that the sample contains all balls of exactly one color
(that is, all the four balls of one color only are among the 8 balls)

32. I have n keys only one of which opens a door. If I try the keys successively
(one key after another), what is the probability that I will find the right
0

key at the last attempt?

33. An urn contains B black balls and W white balls. n balls are drawn
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from the urn at random, their color unnoticed, then m additional balls
are drawn. What is the probability that there are k black balls among the
m balls?
34. A box contains r red balls, w white balls, and b blue balls. Suppose
that balls are drawn from the box one at a time, at random, without

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replacement. What is the probability that all r red balls will be obtained
before any white balls are obtained?
35. Given thirty-three people, find the probability that among the twelve
months there are 3 that contain two birthdays, and 9 that contain three?
Assume the probability of being born in each month is the same for all
months and people.
36. Cards are drawn one by one from a regular deck (13 cards for each of the
4 suits). If 7 cards are drawn, what is the probability that no suit will be
missing?

37. A lecturer has prepared 8 different problems to distribute to her 4 students.

She will do so randomly. What is the probability that all students will
have at least one problem to work on?
38. After assigning the problems, the lecturer in the exercise above distributes
30 white sheets (scrap paper) to her four students. Assume that all possi-
ble distributions are equiprobable. What is the probability that no student
will have fewer than 3 sheets?
39. Prove combinatorially that
  k   
n−1+k X n k−1
=
k i=1
i k−i

Hint: the LHS (left hand side) counts the number of ways in which we
can place k indistinguishable objects in n labelled boxes with no limit on
how many balls can be had in a box. To compute the RHS, count all the
ways of allocating all balls in one box, all the ways of allocating all balls
into two boxes (filling both boxes), etc
40. Petra is organizing her birthday party. She needs to buy some refreshments
and calls a local distributor to have some soda delivered. The distributor
has a special offer for boxfuls of 22 bottles of soda of 5 brands: Pepsi,
Coca Cola, Fanta, Crush, Seven-Up. Eager to take up the offer, Petra
forgets to specify her choice of brands. The distributor delivers one such
boxful randomly selected among all possible different ones. What is the
probability that the boxful has at least 10 bottles of Cola (Pepsi or Coca
Cola), and at least two bottles of each of the other soda brands?
0

A [ ] 0.0914
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B [ ] 0.1072
C [ ] 0.1545
D [ ] 0.0702