A certain club has 50 people, and 4 members are running for president. Each club member either votes for one of the 4 candidates, or can abstain from voting. How many different possible vote totals are there?
I tried approaching it like this:
Using the Hockey-Stick theorem, there are C(53, 3) = 23,426 ways to vote IF they aren't allowed to abstain.
If no one abstains, there are C(53, 3) possible totals.
If one person abstains, there is one person less, so there are then C(52, 3) ways to vote.
This keeps going on like this: C(53, 3) + C(52, 3) + C(51, 3)... all the way to some number I don't know what it would be. Any way to figure this out?