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?

Rationale

C(12, 3) × C(8, 2)

To form a committee of 5 members consisting of exactly 3 juniors and 2 seniors, we must choose 3 juniors from a pool of 12 and 2 seniors from a pool of 8. The total number of different committees is calculated by multiplying the combinations of juniors and seniors, which is represented as C(12, 3) × C(8, 2).

A (C(12, 5)) YOUR ANSWER

This option suggests selecting 5 juniors from the 12 available. However, this fails to account for the requirement of including 2 seniors in each committee, thus violating the specified composition of the group.

B (C(8, 5)) YOUR ANSWER

This choice implies selecting 5 seniors from the 8 available. Since the committee must consist of juniors as well, this option disregards the necessity of having juniors in the group and is therefore incorrect.

C (C(12, 2)) YOUR ANSWER

Choosing 2 juniors from the 12 available does not meet the requirement for the committee composition of 3 juniors and 2 seniors. This selection does not fulfill the criteria set by the problem statement.

D (C(8, 2)) YOUR ANSWER

While this option correctly calculates the number of ways to choose 3 juniors from the 12, it neglects the need to include 2 seniors, making the committee incomplete according to the requirements.

E (C(8, 2)) YOUR ANSWER

This option selects 2 seniors from the 8 available but does not include the 3 juniors needed for the committee, thus failing to meet the specified composition requirements.

F (C(12, 3) × C(8, 2)) CORRECT

This choice is actually the correct answer, but it is listed among the incorrect options due to a mislabeling in the question. It represents the correct computation for forming the specified committee.

G (C(20, 5)) YOUR ANSWER

This option incorrectly suggests selecting 5 members from the total of 20 students (12 juniors and 8 seniors) without regard to the specific junior-senior composition required for the committee. Therefore, it does not satisfy the problem's conditions.

Conclusion

To form a committee of 5 members, including 3 juniors and 2 seniors, the correct method involves calculating C(12, 3) for the juniors and C(8, 2) for the seniors, yielding the overall expression C(12, 3) × C(8, 2). All other options either miscount or ignore the necessary proportions, underscoring the importance of adhering to specified group compositions in combinatorial problems.

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?

C(12, 3) × C(8, 2)

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