Final answer:
To calculate the number of ways to pick a committee of 6, we can use combinations. For each part of the question, we apply the appropriate combinations formula to calculate the different possibilities.
Step-by-step explanation:
To calculate the number of ways to pick a committee of 6, we can use combinations. In this question, we have 12 women and 10 men. Let's solve the different parts of the question:
(a) To ensure that Ann and Beth will not serve together, we subtract the number of ways they can be together from the total number of ways to pick a committee of 6. The number of ways they can be together is the product of the ways to select them (2) and the ways to select the remaining 4 members from the remaining 18 (C(18,4)). So, the number of ways to not have Ann and Beth together is C(22,6) - 2 * C(18,4).
(b) To have at least one woman, we can subtract the number of all-male committees from the total number of committees. The number of all-male committees is the combinations of selecting 6 members from the 10 men (C(10,6)). So, the number of committees with at least one woman is C(22,6) - C(10,6).
(c) To have at least one man and one woman, we can calculate the number of committees with no men or no women and subtract it from the total number of committees. The number of committees with no men is the combinations of selecting 6 members from the 12 women (C(12,6)), and the number of committees with no women is the combinations of selecting 6 members from the 10 men (C(10,6)). So, the number of committees with at least one man and one woman is C(22,6) - C(12,6) - C(10,6).