Assume that the committee consists of 8 Republicans and 6 Democrats. A subcommittee consisting of 7 people is to be selected. How many such subcommittees are possible if each subcommittee must contain at least 1 and no more than 3 Democrats?
To calculate the number of such subcommittees, we need to consider two separate cases: when the subcommittee contains exactly 1 Democrat and when the subcommittee contains exactly 2 or 3 Democrats.
Case 1: Subcommittee with exactly 1 Democrat.
In this case, we choose 1 Democrat out of 6 and 6 Republicans out of 8. The number of ways to select the subcommittee is given by the product of these combinations:
(6C1) * (8C6) = 6 * 28 = 168
Case 2: Subcommittee with exactly 2 or 3 Democrats.
In this case, we choose 2 Democrats out of 6 and 4 Republicans out of 8, or we choose 3 Democrats out of 6 and 4 Republicans out of 8. The number of ways to select the subcommittee is given by the sum of these combinations:
[(6C2) * (8C4)] + [(6C3) * (8C4)] = (15 * 70) + (20 * 70) = 1050 + 1400 = 2450
So, the total number of possible subcommittees is the sum of the two cases:
168 + 2450 = 2618
Therefore, there are 2618 possible subcommittees that can be selected.
To find the number of possible subcommittees, we need to consider the two possibilities separately: one with 1 Democrat, one with 2 Democrats, and one with 3 Democrats.
1. Subcommittees with 1 Democrat:
- Select 1 Democrat from the 6 available Democrats.
- Select the remaining 6 members from the remaining 14 people (8 Republicans + 5 Democrats).
- The number of subcommittees with 1 Democrat is (6 C 1) * (14 C 6).
2. Subcommittees with 2 Democrats:
- Select 2 Democrats from the 6 available Democrats.
- Select the remaining 5 members from the remaining 14 people (8 Republicans + 5 Democrats).
- The number of subcommittees with 2 Democrats is (6 C 2) * (14 C 5).
3. Subcommittees with 3 Democrats:
- Select 3 Democrats from the 6 available Democrats.
- Select the remaining 4 members from the remaining 14 people (8 Republicans + 5 Democrats).
- The number of subcommittees with 3 Democrats is (6 C 3) * (14 C 4).
To find the total number of possible subcommittees, we sum up the number of subcommittees from each case:
Total number of subcommittees = (6 C 1) * (14 C 6) + (6 C 2) * (14 C 5) + (6 C 3) * (14 C 4)
Now, we can calculate the answer using the combinatorial formula.
Do it like the other question I just showed you.
This time I see 3 cases:
1. one Democrat
2. two Democrats
3. three Democrats
I'll do #2
no. of selections = C(6,2) * C(8,5) = 15*56 = 840