A Senate committee has 8 Republicans and 6 Democrats. In how many ways can we form a subcommittee of 5 members that has at least one member from each party?
+235 Seeing how we have low numbers, it's just trial and error. RRRRD, RRRDD, RRDDD, RDDDD, where R is Republicans and D is Democrats. It doesn't matter whether you do RRDRR or RRRRD, they are the same thing. So counting them, we have 4 possible scenarios.
Hope this helps!
+129907 Total possible subcommittees = C(14,5)
Total possible subcommittees that contain no Repubs = C(6,5) [choose any 5 Dems from 6 ]
Total possible subcommittees that contain no Dems = C(8,5) [ choose any 5 Repubs from 8 ]
So...the total number of subcommittees that contain at least one member of either party =
C (14,5) - C(6,5) - C(8,5) = 1940
