Question

How many subsets of five elements can be formed from a set of 80 elements?

Answer 1

`""^80 C_5 = 24040016`

Explanation:

If we were to pick the `5` elements one at a time, then there would be `80` ways to pick the first, `79` to pick the second, `78` to pick the third, `77` to pick the fourth and `76` to pick the fifth.

That is: `""^80 P_5 = (80!)/((80-5)!) = (80!)/(75!)= 80*79*78*77*76`

But the order in which we pick a particular subset does not matter, so we can divide this value by the number of ways of ordering `5` items, that is `5!`, to get:

`""^80C_5 = (80!)/(75!5!) = (80*79*78*77*76)/(5*4*3*2*1) = 24040016`