What if horse order doesn’t matter

So far we’ve found the number of permutations of ordering three horses taken from a group of twenty. This means that we know how many exact arrangements we can make.

This time around, we don’t want to know how many different permutations there are. We want to know the number of combinations of the top three horses instead. We still want to know how many ways there are of filling the top three positions, but this time the exact arrangement doesn’t matter.

So how can we solve this sort of problem?

At the moment, the number of permutations includes the number of ways of arranging the 3 horses that are in the top three. There are 3! ways of arranging each set of 3 horses, so let’s divide the number of permutations by 3!. This will give us the number of ways in which the top three positions can be filled but without the exact order mattering.