They have changed the rules for resolution passage within the UN Security Council. It still takes all 5 permanent members to vote “for” a resolution, but it will now take a total of 11 “for” votes to pass. How many ways can a resolution pass?

Answer 1

It depends on how many members there are in the council. If there are only 11, then only a unanimous vote will do.

If there are N members, then at least 6 of the N-5 members will need to vote yes.

Answer 2

There are 10 non-permanent and 5 permanent members for a total of 15.

This means that there must be at least n votes from the 10 non-permanent members, where n≥6.

So the number of ways the votes can go are:
C(10,6)+C(10,7)+C(10,8)+C(10,9)+C(10,10)
=386
where C(10,6)=10!/(6!(10-6))
(check my arithmetic).