Basics of Combinations

First make sure to read the basics of permutations. So what’s the different between permutations and combinations?

Suppose we have set A = {a,b,c,d,e}. A permutation of that set could be abc, and another permutation could be acb. In other words, a permutation is an arrangement of the objects of set A, where order matters.

A combination of set A, on the other hand, is simply a subset of that set. So both abc and acb would be considered the same combination.

So how do we find the total number of combinations?

First consider that it’s possible to find the total number of permutations based on the number of combinations. That is, the total number of permutations of a set with n elements where you want to permute k at a time is equal to the total number of combinations of that set (again choosing k at a time) multiplied by the number of ways you can permute the elements of each combination, which is basically k!. So:

P = C * k!

Moving k! to the other side we have:

C = P / k!

By expanding the P formula we have:

C = n! / k! (n-k)!

Suppose you have set A = {a,b,c,d,e} and you want to permute the elements 3 at a time. So n=5 and k=3. The total number of combinations would be 10, so the total number of permutations is 10 * 3!, which is 60.