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Math Help - how many ways can 8 boys be divided?

  1. #1
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    how many ways can 8 boys be divided?

    how many ways can 8 boys be divided into two unequal sets?

    the answer said 92. But i dont get it!!!

    Thanks
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  2. #2
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    Quote Originally Posted by differentiate View Post
    how many ways can 8 boys be divided into two unequal sets?

    the answer said 92. But i dont get it!!!

    Thanks
    Hi differentiate,

    To divide 8 boys into two unequal sets,

    they may be divided into

    (a) groups of 1 and 7,
    (b) groups of 2 and 6,
    (c) groups of 3 and 5

    but not groups of 4 and 4 as they would be equal sets
    and groups of 0 and 8 is one set.

    There are \binom{8}{1}=8_{C_1} ways to divide into groups of 1 and 7,
    since that's the number of ways to choose 1 of the 8,
    (leaving 7 in the other group).

    There are \binom{8}{2}=8_{C_2} ways to divide into groups of 2 and 6,
    since that's the number of ways to choose 2 of the 8,
    (leaving 6 in the other group).

    There are \binom{8}{3}=8_{C_3} ways to divide into groups of 3 and 5,
    since that's the number of ways to choose 3 of the 8,
    (leaving 5 in the other group).

    Remember

    \binom{8}{1}=\binom{8}{7}

    \binom{8}{2}=\binom{8}{6}

    \binom{8}{3}=\binom{8}{5}

    (the number of ways to choose 1 from 8) = (the number of ways to choose 7 from 8)

    (the number of ways to choose 2 from 8) = (the number of ways to choose 6 from 8)

    (the number of ways to choose 3 from 8) = (the number of ways to choose 5 from 8)
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