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June 2022 p51 q2
2881
There are 6 men and 8 women in a Book Club. The committee of the club consists of five of its members. Mr Lan and Mrs Lan are members of the club.
(a) In how many different ways can the committee be selected if exactly one of Mr Lan and Mrs Lan must be on the committee?
(b) In how many different ways can the committee be selected if Mrs Lan must be on the committee and there must be more women than men on the committee?
Solution
(a) To find the number of ways to select the committee with exactly one of Mr Lan or Mrs Lan, we consider two cases:
Mr Lan is on the committee, but Mrs Lan is not. We choose 4 more members from the remaining 12 people (6 men and 6 women): \\(\binom{12}{4} = 495 \\\)
Mrs Lan is on the committee, but Mr Lan is not. We choose 4 more members from the remaining 12 people (6 men and 6 women): \\(\binom{12}{4} = 495 \\\)
Thus, the total number of ways is \\(495 + 495 = 990 \\\).
(b) If Mrs Lan must be on the committee and there must be more women than men, we consider the following scenarios:
2 women and 2 men in addition to Mrs Lan: \\(\binom{7}{2} \times \binom{6}{2} = 315 \\\)
3 women and 1 man in addition to Mrs Lan: \\(\binom{7}{3} \times \binom{6}{1} = 210 \\\)
4 women in addition to Mrs Lan: \\(\binom{7}{4} = 35 \\\)
Adding these gives the total number of ways: \\(315 + 210 + 35 = 560 \\\).