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24 April, 07:37

How many ways can a group of 12 fraternity brothers be assigned to rooms a, b, and c, with 4 assigned to room a, 6 to room b, and 2 to room c?

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  1. 24 April, 11:00
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    Assign 4 brothers to room A in C (12,4) = 12! / (4!8!) ways

    Assign 6 brothers to room B in C (8,6) = 8! / (6!2!) ways

    Assign 2 brothers to room C in C (2,2) = 2! (2!0!) ways

    Total number of ways is the product

    C (12,4) * C (8,6) * C (2,2) = 12!8!2! / (4!8!6!2!2!0!) = 12! / (4!6!2!) = 13860 ways

    Incidentally, this is the same answer as the number of permutations of arranging 12 objects in a line, with 6,4 and 2 objects being identical

    P (4,6,2) = 12! / (4!6!2!) = 13860.

    Think about how this relates to the problems of the brothers.
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