Chapter 6 Counting,
Permutations and Combinations
Name:_________________
Warm Up:
Suppose you and a friend went to a pet store. You were deciding if you wanted to buy a dog, cat, or
hamster. They have each animal with black or brown fur. How many total options do you have for pets?
What if they had each animal in white, brown, or black?
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Key Concept: Fundamental Counting Principle
If one event can occur in m ways and another event can occur in n ways, then the number of ways both
events could occur is ___________.
This principle can be extended to three or more events. For example, if three events can occur in m, n,
and p ways, then the number of ways that all three events can occur is ________________.
The standard configuration for a license plate is 1 letter followed by 2 digits followed by 3 letters.
How many different license plates are possible if letters and digits can be repeated?
What about if you can’t use the same letter/number more than once?
If I’m trying to take a photo of Jessie, Joe, Militsa and Tayseer all standing in a row how many ways could
I organize the photo?
Questions to consider: How many options do I have for the first person in line? How many options, then,
for the second person? Remember the fundamental counting principle.
Definition: Factorial – 𝑛! = 𝑛 ∙ (𝑛 − 1) ∙ (𝑛 − 2) ∙ … ∙ 3 ∙ 2 ∙ 1
Suppose there are 10 teams competing in a round of the world cup. How many different ways could the
soccer teams finish in the competition (if there are no ties).
How many ways could these teams finish first, second, and third in order to medal?
Key Concepts: Permutations of n Objects Taken r at a Time
The number of permutations of r objects taken from a group of n distinct objects is denoted by nPr and is
given by the following formula:
nPr=
𝒏!
(𝒏−𝒓)!
Example: You’ve got 15 students running for five different class officer spots. How many different
combinations of class officers could you have?
How many ways can you arrange the letters in the word HELLO?
Permutations with Repetition:
The number of distinguishable permutations of n objects where one object is repeated s1 times, another
is repeated s2 times, and so on, is:
𝑛!
𝑠1 ! ∙ 𝑠2 ! ∙ … ∙ 𝑠𝑘 !
Example: How many ways can you arrange the letters in PANAMA?
Now consider you have 15 students applying to be on a class council that has 5 spots. No position on the
council is any different than the other. Does it matter the order in which the candidates are selected?
For example: Is a committee of Emily, Ahmed, Sraavani, Kevin, and Kathleen different than a committee
made up by Ahmed, Emily, Kevin, Sraavani and Kathleen?
This is called a combination. A combination can be solved using the formula:
nCr
𝑛!
= (𝑛−𝑟)!∙
𝑟!
Example 1: 15 students are trying out for 5 open spots on a basketball team. How many different
combinations are there for the team?
Example 2: There are 14 types of toppings on a Domino’s Pizza. For $5 you may pick 3 different toppings.
How many combinations are there?
Practice:
Find 8C3
7P4
Discuss the difference between Permutations and Combinations:
9C3