Combinations and Permutations

Combinations and permutations are the number of ways you can arrange or group things. If the order doesn't matter when you arrange something, then it is called a combination. When order does matter, it is called a permutation.

When a problem asks the question "how many ways" it is probably a combination problem.


Combinations

Here is an example of a combination: suppose 5 people are playing a game where they win if they are in a chair when the music stops. But there are only two chairs. Because it doesn't matter which chair they end up sitting in, this is a combination problem.

The question is really: How many ways can we pick the two people who get a chair?

One method is to make a list - let's call the people April, Bob, Chuck, Debbie and Eric, or A, B, C, D, and E for short. Here is our list:

Order doesn't matter
AB AC AD AE
BC BD BE
CD CE
DE
10 ways
combination

Since it doesn't really matter if you get in a chair first or second (just that you get a chair) then this is an "order doesn't matter" or combination problem and the answer is 10 ways.


Permutation

When "order does matter" when you are picking things, then you have a permutation problem

If we change our game so that the person who gets the first chair gets 1st place and the person who gets the second chair gets 2nd place in our game, then the order does matter and this is a permutation problem.

Let us try to solve the problem by counting all the possibilities again. We list all the different ways that A can be in the first chair on the first row. On the second row, we list all the ways that B can be in the first chair. Continuing until each player has a chance to be in the first chair, we have:

Order does matter
AB AC AD AE
BA BC BD BE
CA CB CD CE
DA DB DC DE
EA EB EC ED

20 ways

permutation

By counting all the possible ways, we have found that the answer is 20 ways.

Here is another way to find the permutation answer using the fundamental counting principle and a little bit of math logic. Since there are 5 people and any one of them can get the first chair, there are 5 ways to get in the first chair. Now, since there are only 4 people left to fight over the second chair, there are 4 ways to get in the second chair.

possible ways to win = 5 choices in 1st chair   4 choices in 2nd chair = 20 ways

Wouldn't it be nice if we could find a shortcut method this simple for combinations? Well we can by using factorials: the ways when order doesn't matter equals the number of choices total (permutations) divided by the number being picked (n!). We found out that we had 20 total ways and there are only 2 chairs (2! = 2 1 = 2), so the combinations are:

20 ways  =  10 combinations
2 choices

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Last Updated: 8/1/2007
K. A. Shearer 2007
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