"a club with 20 members must choose a three-person committee and a five-person committee. how many ways can the two committees be chosen if the committees can overlap? how many ways can the two committees be chosen if the committees cannot overlap?".
To calculate the number of ways the committees can be chosen, we use the concept of combinations.
1. Committees can overlap:
To form a three-person committee, we need to choose 3 members from a total of 20 members. This can be done using a combination formula, denoted as "nCr," where n represents the total number of members and r represents the number of members to be chosen for the committee. In this case, n = 20 and r = 3. So, the number of ways to form the three-person committee is:
20C3 = 20! / (3! * (20-3)!) = 1140
Similarly, to form a five-person committee, we need to choose 5 members from a total of 20 members. Using the combination formula, we get:
20C5 = 20! / (5! * (20-5)!) = 15504
Since the committees can overlap, the number of ways to choose both committees is obtained by multiplying the number of ways to choose each committee:
Number of ways = 1140 * 15504 = 17,731,760
2. Committees cannot overlap:
If the committees cannot overlap, we need to consider that once a member is chosen for one committee, they cannot be chosen for the other committee. In this case, we use the concept of combinations with restrictions.
To form a three-person committee, we choose 3 members from a total of 20 members without replacement. The number of ways to do this is similar to the calculation above:
20C3 = 1140
However, for the five-person committee, we need to choose 5 members from the remaining 17 members after the three-person committee is formed. So, the number of ways to form the five-person committee is:
17C5 = 17! / (5! * (17-5)!) = 6188
Now, to calculate the number of ways to choose both committees, we multiply the number of ways to choose each committee:
Number of ways = 1140 * 6188 = 7,056,720
To summarize:
1. Committees can overlap: 17,731,760 ways
2. Committees cannot overlap: 7,056,720 ways