Assume that the committee consists of 4 Republicans and 5 Democrats. A subcommittee consisting of 5 people is to be selected.
How many such subcommittees are possible if each subcommittee must contain at least 1 and no more than 2 Republicans?
Well, selecting a subcommittee can be quite political, but let's calculate the possibilities without causing too much political uproar.
To form a subcommittee, we need to consider the number of Republicans and Democrats. We have 4 Republicans and 5 Democrats in the committee.
Let's break it down into cases:
Case 1: Selecting 1 Republican:
There are 4 ways to choose 1 Republican and 5 Democrats to complete the subcommittee. So, there are 4 subcommittees in this case.
Case 2: Selecting 2 Republicans:
There are 4 Republicans to choose from, and we need to select 2. So, we have 4C2 = 6 ways to choose 2 Republicans. For the remaining members, we have 5 Democrats to choose from, and we need to select 3. So, we have 5C3 = 10 ways to choose 3 Democrats. Therefore, there are 6 x 10 = 60 subcommittees in this case.
Adding up the subcommittees from both cases gives us 4 + 60 = 64 possible subcommittees.
Remember, these numbers are given assuming that politics won't get in the way and everyone works together like a well-oiled, non-partisan machine.
To find the number of possible subcommittees, we need to consider two cases:
Case 1: 1 Republican and 4 Democrats
In this case, we choose 1 Republican out of 4 and 4 Democrats out of 5. The total number of subcommittees with this combination is given by the product of the number of ways to choose the Republicans and the number of ways to choose the Democrats:
Number of subcommittees = C(4, 1) * C(5, 4) = 4 * 5 = 20
Case 2: 2 Republicans and 3 Democrats
In this case, we choose 2 Republicans out of 4 and 3 Democrats out of 5. The total number of subcommittees with this combination is given by the product of the number of ways to choose the Republicans and the number of ways to choose the Democrats:
Number of subcommittees = C(4, 2) * C(5, 3) = 6 * 10 = 60
Therefore, the total number of possible subcommittees is the sum of the number of subcommittees from case 1 and case 2:
Total number of subcommittees = 20 + 60 = 80.
So, there are 80 possible subcommittees.
To find the number of possible subcommittees, we can consider the different possible cases for the number of Republicans in the subcommittee.
Case 1: 1 Republican and 4 Democrats
In this case, we have to choose 1 Republican out of 4 available Republicans and 4 Democrats out of 5 available Democrats. So, the number of subcommittees with 1 Republican and 4 Democrats is:
C(4, 1) * C(5, 4) = 4 * 5 = 20
Case 2: 2 Republicans and 3 Democrats
In this case, we have to choose 2 Republicans out of 4 available Republicans and 3 Democrats out of 5 available Democrats. So, the number of subcommittees with 2 Republicans and 3 Democrats is:
C(4, 2) * C(5, 3) = 6 * 10 = 60
Therefore, the total number of subcommittees that can be formed with at least 1 and no more than 2 Republicans is the sum of subcommittees from Case 1 and Case 2:
20 + 60 = 80
Hence, there are 80 possible subcommittees that can be formed.
If C(M,N) represents the number of combinations by choosing N objects out of M, and equals M!/((M-N)!N!)
Case 1:
1 republican + 4 democrats
P(1,4)
=C(4,1)*C(5,4)
=4!/(1!3!) * (5!/(4!1!))
=20
Case 2:
2 republicans + 3 democrats
P(2,3)
=C(4,2)*C(5,3)
=4!/(2!2!) * (5!/(3!2!))
=12*10
=120
Total number of different committees
=20+120
=140