In this example, can somebody explain the middle section? Why is it 8 choose 2? Thanks!
What do you think it should be?
10 choose 2?
You need to pick a pool from 8 (not 10), because we removed Bob and Bill from the group of 10.
So in that combinatorics, we are essentially assuming Bob and Bill are given, and then figuring how many other pairs are possible giving the remaining people?
Imagine the following scenario. From \{A,B,C\}, if you pick A, then how many partners can you pair up with A? It should just be the number of singletons you can make from \{B,C\}, right? The same logic extends to the question at hand.
From an initial set of 10 seniors, we picked out Bob and Bill. Now we need 6 others to accompany Bob and Bill from the remaining pool of 8. As such, we’re counting how many 6-tuples we can make from 8, which is \binom{8}{6} = \binom{8}{2}.
Yes, in a way.
Great, thank you for clarifying!
Thanks!