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24 February, 22:03

8. In how many ways can a committee of four be selected from eight men and seven women if Miss Jones refuses to serve on the same committee as Mr. Smith?

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  1. 24 February, 22:28
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    If the exclusion of Miss Jones and Mr Smith serving together was not present, the are 15C4 = 1365 ways of selecting the committee.

    Miss Jones can serve on the committee in the following ways:

    a) with 3 men

    b) with 2 men and another woman

    c) with 1 man and 2 other women

    d) with 3 other women.

    Arrangement d) obviously presents no restrictions.

    Arrangement a) has 7C3 ways excluding Mr Smith, and 8C3 ways if Mr Smith was included.

    Arrangement b) has 7C2 * 6 ways excluding Mr Smith, and 8C2 * 6 ways if Mr Smith was included.

    Arrangement c) has 7 * 6C2 ways excluding Mr Smith, and 8 * 6C2 ways if Mr smith was included.

    The reductions in ways caused by the restriction are as follows:

    a) 8C3 - 7C3 = 21 ways

    b) 6 (8C2 - 7C2) = 42 ways

    c) 6C2 = 15 ways

    The total reduction in the number of ways is: 21 + 42 + 15 = 78.

    Therefore the total number of ways of selecting the committee, while observing the restriction, is 1365 - 78 = 1287 ways.
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