Final answer:
To form committees with specific numbers of prefects from a class of 12 girls, we use combinations. The correct answer for forming committees with exactly 3 prefects and with at least 3 prefects involve calculating the combinations of prefects and non-prefects to fill the required committee spots.
Step-by-step explanation:
In a situation where a class consists of 12 girls, with 5 of them being prefects, and we want to form committees of 8, the approach to solve this depends on the specific requirements for prefects on the committee.
(a) Committees with exactly 3 prefects:
We use combinations to select the committee members. There will be C(5,3) ways to choose the 3 prefects from the 5 available, and C(7,5) ways to choose the remaining 5 members from the non-prefects (since there are 12 - 5 = 7 non-prefect girls).
The total number of committees that can be formed with exactly 3 prefects is therefore C(5,3) * C(7,5).
(b) Committees with at least 3 prefects:
For committees with at least 3 prefects, we consider committees with 3, 4, or all 5 prefects. This is the sum of:
- C(5,3) * C(7,5) for 3 prefects,
- C(5,4) * C(7,4) for 4 prefects, and
- C(5,5) * C(7,3) for all 5 prefects.
The total number of committees that can be formed with at least 3 prefects is C(5,3) * C(7,5) + C(5,4) * C(7,4) + C(5,5) * C(7,3).
The correct answer is option A: (a) C(5,3) * C(7,5), (b) C(5,3) * C(7,5) + C(5,4) * C(7,4) + C(5,5) * C(7,3).