Combinations

Last updated: 20 Mar 2009

This article originally appeared on PlainMath.net and is reprinted with their kind permission

Unlike a permutation, in a combination the order of the objects selected does not matter. For example, if you were choosing a team of three students from a class of ten, the order you said "Joe, Sally, John" would not matter to who is in the group- saying "Sally, Joe, John" instead doesn't change anything.

How to solve a combination

The number of ways you can make a combination of r objects out of a set of n objects is made from the formula below:

$mathematical expression$

An explanation

You may be asking (like I did when I learned this material), "What? That formula doesn't make sense!". However, it is based on the formula for a permutation.

If you think about it, the $mathematical expression$ makes perfect sense - that is how many options you'd have if you picked in order from all of them.

Removing the $mathematical expression$ gets it down to just the numbers you want, where $mathematical expression$ equals the last number you aren't picking. If you pick 3 from 5, it'll be

$mathematical expression$

Which accomplishes the "pick x of 5" part.

The extra $mathematical expression$ removes all the redundant options. For example, say you have 6 different balls, labelled A through F. If you pick three, you can pick ABC, ABD, ABE, ABF, etc. You'll end up with six of the same — ABC, ACB, BAC, BCA, CAB, CBA — all of which only count as one in a combination. As you'll notice, since we picked 3, $mathematical expression$ - the same number we have to divide out!

An example of a combination

Lets take the example above. Out of a class of ten students, how many ways could you make a team of three students? We know $mathematical expression$ is ten because the set (the students) has ten objects in it. We know $mathematical expression$ is 3 because the team will have three students on it. To solve, we use the combinations equation.