Chapter 11. Section 3
Page 1
Section 11.3 – Combinations
Homework (pg 599) problems 1-38
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Definition: A combination problem involves situations when items are selected from the same
group, no items is used more than once and the order of the items selected doesn’t matter.
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Example: If I choose three students to take a survey, then it doesn’t matter how I select them, just
that I pick three. That makes it a combination problem.
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Example: Recall that a permutation is similar, but order does matter. Instead of giving the same
survey to all three students, say I want to give three different surveys to three students. Here, order
does matter because each student could take one of three different surveys
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Example (Checkpoint 1): Explain if the problems are permutations or combinations
a) How many ways can you select 6 free videos from a list of 200?
b) In a race where there are 50 runners and no ties, in how many ways can the first three finishers
come in?
Solution:
a) You are not ranking the videos, only picking them, so order does not make a difference.
This is an example of a combination
b) Here order does make a difference, because there is a first, second and third place. Order
matters here, so this is an example of a permutation
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The number of possible combinations if r itmes are taken from n items is given by the formula
n!
n Cr =
( n − r )! r !
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Example (Checkpoint 2): You volunteer to pet-sit for your friend who has seven different animals.
You offer to take three of the seven. How many different pet combinations can you care for?
Solution:
Here, order does not make a difference, so it is a combination.
You are choosing three out of seven
7!
= 35
7 C3 =
(7 − 3)! 3!
Chapter 11. Section 3
Page 2
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Question: What is the relationship between permutations and combinations of the same set? Say
you are choosing 2 items out of a group of 6.
Which will be a larger value of the same set, a permutation or a combination?
A permutation will be larger
6!
6!
= 30
= 15
6 P2 =
6 C2 =
(6 − 2)!
(6 − 2)! 2!
If you are choosing 2 items, how many ways can they be arranged?
2. A and B, or B and A
In general, if you are choosing n items they can be arranged in n! different ways.
That is the difference between the formulas. In a combination the order doesn’t matter, so the
combination formula divides by the number of possible arrangements.
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Example (Checkpoint 3): How many different 4-card hands can be dealt from a deck that has 16
different cards?
Solution: Again, order doesn’t matter. You are choosing 4 out of 52
52!
= 270,725
52 C 4 =
(52 − 4)! 4!
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Example (Checkpoint 4): The US Senate of the 107th Congress consisted of 50 Democrats, 49
Republicans and one Independent. How many committees can be formed if each committee must
have 3 Democrats and 2 Republicans?
Solution:
The question would be better worded as, “how many ways can you select a committee of 3
democrats and 2 republicans”
These are two exclusive groups.
50!
We can choose the democrats in 50 C3 =
= 19,600 ways
(50 − 3)! 3!
49!
We can choose the republicans in 49 C2 =
= 1,176 ways
(49 − 2)! 2!
All together, we can choose the committee in 19600(1176) = 23,049,600 ways