Counting Combinations

Just as with permutations, it’s sometimes easiest to calculate combinations by listing all the possibilities available, and counting them. The only difference is, when we list one combination, we automatically exclude a larger number of permutations. For example a poker hand that is (ace, ace, ace, ace, king♣) is identical to (ace, ace, ace, king♣, ace), (ace, ace, king♣, ace, ace), (ace, king♣, ace, ace, ace) and (king♣, ace, ace, ace, ace). So we must be careful to use a listing method that includes all combinations without repeating ones that are really the same. Let’s examine a situation where it’s relatively straightforward to do that.

Anne wishes to knit herself a striped sweater. She has 4 colors of yarn available; red, blue, green and yellow. How many different combinations of two colors does she have to choose from?

When we just choose color pairs, there will be fewer combinations than we would have if we were counting permutations as in the previous lesson. For example red and blue is equivalent to blue and red, and we should only count one as a unique pairing. We start by listing the color pairs but we will also write down equivalent pairings at the same time. This will help prevent us from repeating combinations:

So there are 6 distinct combinations. There are also 6 “repeat” pairings – for every pair of colors we choose there is 1 combination but 2 permutations. Anne can choose from six distinct color pairs for her sweater.