in the class must appoint three corporate leaders; the class president ,the vice president and the entertainment director.it must also appoint a planning commitee with different members .there are 8 qualified candidate , and lenders can also serve on the planning committee. a. how many different ways can the Leader be appointed? b.how may different can the committee be appointed ?
a. For the three corporate leader positions (class president, vice president, and entertainment director), there are 8 candidates to choose from. Since each position must be filled by a different candidate, the first position can be filled by any of the 8 candidates, the second position can be filled by any of the remaining 7 candidates, and the third position can be filled by any of the remaining 6 candidates. Therefore, the number of different ways the corporate leaders can be appointed is:
8 * 7 * 6 = 336
b. For the planning committee, we can choose any number of members, including 0 (in case the committee is not appointed). Since lenders can also serve on the planning committee, we have the following possibilities for the committee size:
1. 0 members (empty committee)
2. 1 member (can be any of the 8 candidates or a lender)
3. 2 members (can be any combination of the 8 candidates and/or lenders; order doesn't matter)
4. 3 members
5. 4 members
6. 5 members
7. 6 members
8. 7 members
9. 8 members (all candidates and lenders included)
Therefore, there are a total of 9 different ways the committee can be appointed.
a. To appoint the three corporate leaders (class president, vice president, and entertainment director) from 8 qualified candidates, we can use the concept of permutations because the order matters in this case.
The number of ways to arrange 3 individuals out of 8 is represented by the permutation formula:
P(8,3) = 8! / (8-3)!
P(8,3) = 8! / 5!
P(8,3) = (8 * 7 * 6) / (3 * 2 * 1)
P(8,3) = 336
Therefore, there are 336 different ways to appoint the three leaders.
b. For the planning committee, we need to consider that the committee members can include both the qualified candidates as well as the lenders. Thus, all 8 qualified candidates can serve on the committee, but the number of lenders is not specified. This means that we have a choice of including or not including each lender on the committee.
Using the concept of combinations, we can calculate the number of committee arrangements:
2^8 = 256
Therefore, there are 256 different ways to appoint the committee, taking into account the possibility of including or excluding each lender.