Combinations at a Glance

A combination is a selection of objects where the order doesn't matter. If you order a combination meal from McDonald's, for example, it doesn't matter if you eat the soda, fries, or burger first. One way or another, all of those calories are getting in there.

If we take a combination of two books from The Iliad, The Odyssey, and Beowulf, the possibilities are:

The Iliad, The Odyssey
The Iliad, Beowulf
The Odyssey, Beowulf

These are the only possible two-book combinations. When we don't care about order, having The Iliad and The Odyssey is the same as having The Odyssey and The Iliad. However, you should really read them in chronological order, or you'll be totally lost. Sirens-what-now?

To find the number of combinations, first we find the number of permutations. Then we divide by the number of ways we can rearrange the permutations. Going with the books again, here are the possible permutations of 2 books out of 3:

The Iliad, The Odyssey
The Odyssey, The Iliad

The Iliad, Beowulf
Beowulf, The Iliad

The Odyssey, Beowulf
Beowulf, The Odyssey

There are 6 permutations, but only half of them count for combinations if we ignore order. Therefore, we have

combinations.

Generalizing this, if we have n objects, take r of them, and don't care about order, there are

possible combinations. These formulas are looking more and more like a cat sat on our keyboard, but trust us: it works.

First we find the number of permutations. Since each permutation can be rewritten in r! different ways, we need to divide by r! for combinations rather than permutations.

Of course, since mathematicians like to abbreviate things, we abbreviate here too. Instead of writing "the number of combinations if we have n objects, take r of them, and don't care about order," we write (C is for combinations). See? Sometimes abbreviations and symbols are our friends. Like when our car conks out on the freeway and they come by to give us a jump.

Here's the combination formula:

Since we also know , we can do a little rewriting:

The formula

is what's usually given for combinations, but it's nice to know where it comes from so that we don't have more "?"s than "!"s.

One neat thing about this formula is that, whether you choose r things or n – r things, you get the same number of combinations.

Sample Problem

Find  and .

Now for the other one:

This is the same as what we got in the second line above, so the answer is 56 here also. Don't you love being able to skip steps? Except when you get a little too cocky heading upstairs, lose your footing, and end up eating the railing?

Our conclusion that the two are equal makes sense, because if we have 8 books and space for 5 on the shelf, we can think of the combinations in two ways. We can think that we have 8 books and need to choose 5 to put on the shelf, or we can think that we have 8 books and need to choose 3 to leave off the shelf. Either way, we find the same possibilities for the books on the shelf.

Remember when we only had two books? It's encouraging to see our library growing. If it gets any larger, people are going to start thinking we actually read these things rather than just put them out for display.

There are a couple more fun things to go over before we leave combinations behind: since there's only one way to choose n out of n objects if order doesn't matter, you can take all of them, which should come as welcome news to all of you possessive people out there:

Also, there's only one way to choose no objects from n, which is to not take any of them:

Here's a handy calculator for checking your work. Please turn it off when you're done.

We suggest that you get comfortable doing factorials by hand, though. You never know when the teacher will decide to give a "no calculators allowed" sort of test. However, such practices are considered discriminatory by certain Texas Instruments products. Calculators aren't going to take it anymore. Too many people have been pushing their buttons for too long.