Assume that the committee consists of 5 Republicans and 7 Democrats. A subcommittee consisting of 6 people is to be selected.
(1) How many such subcommittees are possible if each subcommittee must contain exactly 3 Republicans and 3 Democrats?
(1) How many such subcommittees are possible if each subcommittee must contain at least 1 and no more than 3 Democrats?
(1) To calculate the number of possible subcommittees with exactly 3 Republicans and 3 Democrats, we need to choose 3 Republicans out of 5 and 3 Democrats out of 7. The number of ways to do this is given by the combination formula:
C(5, 3) * C(7, 3) = (5! / (3! * (5-3)!)) * (7! / (3! * (7-3)!))
= (5! / (3! * 2!)) * (7! / (3! * 4!))
= (5 * 4 * 3!) / (3! * 2 * 1) * (7 * 6 * 5!) / (3! * 4 * 1)
= (5 * 4) * (7 * 6) / (2 * 1)
= 20 * 42 / 2
= 840
So, there are 840 possible subcommittees with exactly 3 Republicans and 3 Democrats.
(2) To calculate the number of possible subcommittees with at least 1 and no more than 3 Democrats, we can use the principle of inclusion-exclusion.
We need to find the total number of subcommittees and subtract the ones that have no Democrats and the ones that have more than 3 Democrats.
Total subcommittees = C(12, 6)
= (12! / (6! * (12-6)!))
= (12! / (6! * 6!))
= (12 * 11 * 10 * 9 * 8 * 7 * 6!) / (6! * 6!)
= (12 * 11 * 10 * 9 * 8 * 7) / (6 * 5 * 4 * 3 * 2 * 1)
= 924
Subcommittees with no Democrats = C(5, 6)
= 0
Subcommittees with more than 3 Democrats = C(7, 4) + C(7, 5) + C(7, 6) + C(7, 7)
= 0 + 0 + 0 + 1
= 1
Subcommittees with at least 1 and no more than 3 Democrats = Total subcommittees - Subcommittees with no Democrats - Subcommittees with more than 3 Democrats
= 924 - 0 - 1
= 923
So, there are 923 possible subcommittees with at least 1 and no more than 3 Democrats.
To answer question (1), we need to count the number of possibilities for subcommittees with 3 Republicans and 3 Democrats.
Step 1: Count the number of ways to choose 3 Republicans from the 5 available. This can be done using the combination formula: C(5, 3) = 10.
Step 2: Count the number of ways to choose 3 Democrats from the 7 available. This can be done using the combination formula: C(7, 3) = 35.
Step 3: Multiply the results from Step 1 and Step 2 together to get the total number of possibilities: 10 * 35 = 350.
Therefore, there are 350 possible subcommittees with exactly 3 Republicans and 3 Democrats.
To answer question (2), we need to count the number of possibilities for subcommittees with at least 1 and no more than 3 Democrats.
Step 1: Count the number of possibilities with exactly 1 Democrat:
- Choose 1 Democrat from 7: C(7, 1) = 7
- Choose 5 Republicans from 5: C(5, 5) = 1
- Multiply the results: 7 * 1 = 7 possibilities with exactly 1 Democrat.
Step 2: Count the number of possibilities with exactly 2 Democrats:
- Choose 2 Democrats from 7: C(7, 2) = 21
- Choose 4 Republicans from 5: C(5, 4) = 5
- Multiply the results: 21 * 5 = 105 possibilities with exactly 2 Democrats.
Step 3: Count the number of possibilities with exactly 3 Democrats:
- Choose 3 Democrats from 7: C(7, 3) = 35
- Choose 3 Republicans from 5: C(5, 3) = 10
- Multiply the results: 35 * 10 = 350 possibilities with exactly 3 Democrats.
Step 4: Add up the results from Steps 1, 2, and 3 to get the total number of possibilities: 7 + 105 + 350 = 462.
Therefore, there are 462 possible subcommittees with at least 1 and no more than 3 Democrats.