In the previous two sections we discussed permutations, which are arrangements in which the order of the objects selected matters. For example, If we have a hat with slips of paper numbered from 0 to 9 in it, and we want to select three slips of paper in order to make a three digit number, order matters, because 125 is not the same as 215 or 512.

Sometimes, though, order doesn't matter. Suppose, for instance, that instead of the example above, we are picking the socks we're going to wear today. The order in which they get picked doesn't matter, because we don't care which sock goes on which foot. In other words, picking a blue sock and a black sock is the same as picking a black sock and a blue sock.

Let's try an example. We have 5 plants, and we want to water two of them. In how many ways can we choose the plants? Since it doesn't matter which one we water first, this is a combination rather than a permutation problem. If it was a permutation problem, we'd do ₅P₂ = 20. But since order doesn't matter, we need to divide by two. Why? Because picking the cactus and then picking the ivy is the same as picking the ivy first and then the cactus - we've counted each possibility twice. Thus, we have 10 ways to pick the plants to water.

Suppose we wanted to water three plants instead of two. The total number of permutations is ₅P₃ = = 60. Now we have to ask ourselves, "How many ways are there to arrange three plants?" because that's the number of times we've counted each possibility. You can arrange 3 plants in ₃P₃ ways. = 6. Now we take our total permutations and divide by the number of times each permutation was repeated: = 10.

As with the permutations, we think, "It sure would be nice if we had a formula to help out with this." Well, we do!

If n is the total number of objects, and r is how many of them we're selecting, then the combination formula looks like this:

_nC_r =

It looks a lot like the permutation formula - the only difference is that we're dividing the permutations by n! - which makes sense, since n! is the number of ways of arranging the n objects, and is therefore how many times we've counted each combination.

1.

Explain the difference between a permutation and a combination.

2.

Calculate ₇C₂.

3.

Calculate ₁₀P₂ - ₁₀C₂

4.

If I have 20 books, and I want to place 5 of them on a shelf, without caring about the order in which they are placed, in how many ways can I select the books?

5.

In how many ways can I choose two letters from the word FLOWER, if I don't care about the order?