Permutation and Combinations
Objectives
•Identify whether a situation is a
combination or a permutation
•Count the possibilities
•Test yourself!
A lunch special includes one main item, one
side, and one drink.
How many different meals can you choose if
you pick one main item, one side, and one
drink?
4 x 3 x 3 =36
Benefits of the Fundamental Counting Principle
A sandwich can be made with 3 different types
of bread, 5 different meats, and 2 types of
cheese. How many types of sandwiches can be
made if each sandwich consists of one bread,
one meat, and one cheese.
Bread 1 2 3
Meat 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5
Cheese 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2
There are 30 possible types of sandwiches
(cumbersome)
Benefits of the Fundamental Counting Principle
A sandwich can be made with 3 different types
of bread, 5 different meats, and 2 types of
cheese. How many types of sandwiches can be
made if each sandwich consists of one bread,
one meat, and one cheese.
3 x 5 x 2 = 30
There are 30 possible types of sandwiches.
Recap
A combination is a grouping of outcomes in which
the order does not matter.
A permutation is an arrangement of outcomes in
which the order does matter.
Recap
Combination
Keywords:
Calculator:
No calculator: 5C3 =
Think of an example:
Permutation
Keywords:
Calculator:
No calculator: Arrange 4 items from 6 on a shelf
Think of an example:
Recap
No calculator!
Record your answers on sheet. Show work.
10
C3 = 10
C?
10
C1 =
10
C10 =
10
C0 =
10
C2 =
Discerning between Combinations and Permutations
Tell whether the following situations involve
combinations or permutations. Then give the
number of possible outcomes.
Show all your work!
Question 1
An English test contains five different essay
questions labeled A, B, C, D, and E. You are
supposed to choose 2 to answer. How many
different ways are there to do this?
Question 2
A voicemail system password is 1 letter
followed by a 3-digit number less than 600.
How many different voicemail passwords are
possible?
Question 3
A family of 3 plans to sit in the same row at a
movie theater. How many ways can the family
be seated in 3 seats?
Question 4
Ingrid is stringing 3 different types of beads on
a bracelet. How many ways can she string the
next three beads if they must include one bead
of each type?
Question 5
Nathan wants to order a sandwich with two of
the following ingredients: mushroom, eggplant,
tomato, and avocado. How many different
sandwiches can Nathan choose?
Question 6
A group of 8 swimmers are swimming in a race.
Prizes are given for first, second, and third place.
How many different outcomes can there be?
Question 7
How many different ways can 9 people line
up for a picture?
Question 8
Four people need to be selected from a class
of 15 to help clean up the campus. How many
different ways can the 4 people be chosen?
See how you are getting on….
…. swap sheets with person behind you.
Question 1
An English test contains five different essay
questions labeled A, B, C, D, and E. You are
supposed to choose 2 to answer. How many
different ways are there to do this?
The order of outcomes is not important, so this
situation involves combinations.
5
C2 =10
Question 2
A voicemail system password is 1 letter
followed by a 3-digit number less than 600.
How many different voicemail passwords are
possible if all digits are allowed?
The order of outcomes is important, so this situation
involves permutations.
26 x 6 x 10 x 10 =15600
Question 3
A family of 3 plans to sit in the same row at a
movie theater. How many ways can the family
be seated in 3 seats?
The order of outcomes is important, so this situation
involves permutations.
ABC BAC CAB
ACB BCA CBA
3x2x1=6
Question 4
Ingrid is stringing 3 different types of beads on
a bracelet. How many ways can she string the
next three beads if they must include one bead
of each type?
The order of outcomes is important, so this situation
involves permutations.
3x2x1=6
Question 5
Nathan wants to order a sandwich with two of
the following ingredients: mushroom, eggplant,
tomato, and avocado. How many different
sandwiches can Nathan choose?
The order of outcomes is not important, so this
situation involves combinations.
4
C2 =6
Question 6
A group of 8 swimmers are swimming in a race.
Prizes are given for first, second, and third place.
How many different outcomes can there be?
The order of outcomes is important, so this situation
involves permutations.
8 x 7 x 6 = 336
Question 7
How many different ways can 9 people line
up for a picture?
The order of outcomes is important, so this situation
involves permutations.
9!= 362,880
Question 8
Four people need to be selected from a class
of 15 to help clean up the campus. How many
different ways can the 4 people be chosen?
The order of outcomes is not important, so this
situation involves combinations.
15 Choose 4 = 1365
Return the sheet to the person in front of you along
with their score (out of 16) so far…..
And now for some more…...
Question 9
Four people need to be selected from a class of
15 to help clean up the campus. How many
different ways can the 4 people be chosen, if
the only two girls refuse to help?
Question 10
A basketball team has 12 members who can
play any position. How many different ways
can the coach choose 5 starting players?
Question 11
A basketball team has 12 members who can
play any position. How many different ways
can the coach choose 5 starting players if the
captain MUST play the first half?
Question 12
When ordering a pizza, you can choose 2
toppings from the following: mushrooms,
olives, pepperoni, pineapple, and sausage.
How many different types of pizza can you
order?
Question 13
Nine people in a writing contest are
competing for first, second and third prize.
How many ways can the 3 people be chosen?
Question 14
You are ordering a triple-scoop ice-cream
cone. There are 18 flavours to choose from
and you don’t care which flavor is on the
top, middle, or bottom. How many different
ways can you select a triple-scoop ice-
cream cone?
Question 15
An art gallery has 12 paintings in storage.
They have room to display 4 of them, with
each painting in a different room. How many
possible ways can they display the 4
paintings?
And swap again…...
Question 9
Four people need to be selected from a class of
15 to help clean up the campus. How many
different ways can the 4 people be chosen, if
the only two girls refuse to help?
The order of outcomes is not important, so this
situation involves combinations.
13 Choose 4 = 715
Question 10
A basketball team has 12 members who can
play any position. How many different ways
can the coach choose 5 starting players?
The order of outcomes is not important, so this
situation involves combinations.
12
C5 = 792
Question 11
A basketball team has 12 members who can
play any position. How many different ways
can the coach choose 5 starting players if the
captain MUST play the first half?
The order of outcomes is not important, so this
situation involves combinations.
11
C4 = 330
Question 12
When ordering a pizza, you can choose 2
toppings from the following: mushrooms,
olives, pepperoni, pineapple, and sausage.
How many different types of pizza can you
order?
The order of outcomes is not important, so this
situation involves combinations.
Combinations: 5 choose 2 = 10
Question 13
Nine people in a writing contest are
competing for first, second and third prize.
How many ways can the 3 people be chosen?
The order of outcomes is important, so this situation
involves permutations.
Permutation: 9 x 8 x 7 = 504
Question 14
You are ordering a triple-scoop ice-cream
cone. There are 18 flavours to choose from
and you don’t care which flavor is on the
top, middle, or bottom. How many different
ways can you select a triple-scoop ice-
cream cone?
The order of outcomes is not important, so this
situation involves combinations.
18 choose 3 = 816
Question 15
An art gallery has 12 paintings in storage.
They have room to display 4 of them, with
each painting in a different room. How many
possible ways can they display the 4
additional paintings.
The order of outcomes is important, so this situation
involves permutations.
12 arrange 4 = 11,880
