33. A group of 12 juniors and 8 seniors is to be divided into committees of 5 members, each containing exactly 3 juniors and 2 seniors. How many different committees are possible?
Answer: F
The number of different committees possible is C(12, 3) × C(8, 2).
To form a committee of 5 members that includes exactly 3 juniors and 2 seniors, we calculate the combinations of juniors and seniors separately. Specifically, we select 3 juniors from the 12 available and 2 seniors from the 8 available, which results in the expression C(12, 3) × C(8, 2).
A) C(12, 5)
This option is incorrect because C(12, 5) calculates the number of ways to choose 5 juniors from a group of 12 without considering the requirement for seniors in the committee. The committee must consist of both juniors and seniors.
B) C(8, 5)
This option is also incorrect as C(8, 5) represents the number of ways to choose 5 seniors from a group of 8. Since the committee needs juniors as well, this does not fulfill the requirement of having 3 juniors and 2 seniors.
C) C(12, 2)
This choice is incorrect because C(12, 2) calculates the number of ways to choose 2 juniors from 12. It does not account for the need to select seniors, nor does it meet the committee's composition requirement of 3 juniors.
D) C(8, 2)
This option is incorrect as C(8, 2) counts the ways to choose 2 seniors from a group of 8, but it fails to include any juniors. The committee must consist of both juniors and seniors, making this option inadequate.
E) C(8, 2)
This is a repeat of Option D and is incorrect for the same reasons. It does not contribute to the formation of a committee with the required composition of 3 juniors and 2 seniors.
F) C(12, 3) × C(8, 2)
This is the correct answer, as it accurately calculates the number of ways to select 3 juniors out of 12 and 2 seniors out of 8. This combination meets the requirement of exactly 3 juniors and 2 seniors in each committee.
G) C(20, 5)
This option is incorrect because C(20, 5) calculates the total number of ways to select any 5 members from a combined group of 20 (12 juniors and 8 seniors) without regard to the specific composition of 3 juniors and 2 seniors required for the committee.
Conclusion
The correct answer, C(12, 3) × C(8, 2), effectively determines the number of distinct committees that can be formed with the specified composition. All other options fail as they either do not take into account the necessary distribution of juniors and seniors or miscalculate the group sizes needed for the committee, thus reinforcing the necessity of the correct approach.