Question

Assume that the committee consists of 6 Republicans

Answer 1

Both of these questions rely upon a function to choose a subset of members from a larger set. Let's assume you have M members in the larger set and you want to pick N of them. First, there's M! different ways to arrange the M members. And after arranging them, you can simply pick the 1st N members in line. So we have M! possibilities. But the order of the 1st N people doesn't really matter, and since there's N! different ways to arrange them, let's divide by N!, giving us M!/N!. But the order of the people we didn't pick (M-N) also doesn't matter, so we need to divide by (M-N)! as well. So we get M!/(N!(M-N)!) as the number of possible ways to pick N people out of M people. So let's define a function to represent this P(M,N) = M!/(N!(M-N)!) (1) How many such subcommittees are possible if each subcommittee must contain exactly 3 Republicans and 4 Democrats? We want 3 out of 6 Republicans and 4 out of 8 Democrats. So the number of possible subcommittees is: P(6,3)*P(8,4) = 20*70 = 1400 (2) How many such subcommittees are possible if each subcommittee must contain at least 1 and no more than 3 Republicans? Let's do this problem as the sum of 3 different types of subcommittees. They are 1 Republican and 6 Democrats, 2 Republicans and 5 Democrats, and 3 Republicans and 4 Democrats. So: P(6,1)*P(8,6) + P(6,2)*P(8,5) + P(6,3)*P(8,4) = 6*28 + 24*56 + 20*70 = 168 + 1344 + 1400 = 2912