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Math Help - Combinatorics and Permuation Question

  1. #1
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    Combinatorics and Permuation Question

    I was looking at some problems in a book asking:

    1. How many ways are there to divide a 331 page book into 8 chapters assuming each chapter is an integer number of pages?

    2. Prove that for all n greater than 2, C(n,3) = C(2,2) + C(3,2) + ... + C(n-1,2).


    I've looked at them for awhile but not really sure how to approach it. Any ideas?


    Thanks
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  2. #2
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    Quote Originally Posted by Poplock View Post
    1. How many ways are there to divide a 331 page book into 8 chapters assuming each chapter is an integer number of pages?
    You are asking to solve,
    x_1+x_2+...+x_8 = 331 where x_i \geq 1.
    The number of solutions is {{330}\choose 7}.
    2. Prove that for all n greater than 2, C(n,3) = C(2,2) + C(3,2) + ... + C(n-1,2)
    {k\choose 2} = \frac{1}{2}k^2 - \frac{1}{2}k

    You want to show \sum_{k=2}^{n-1} \frac{1}{2}k^2 - \frac{1}{2}k = {n\choose 3} = \frac{n(n-1)(n-2)}{6}

    Here use the identities, \sum_{k=1}^n k = \frac{k(k+1)}{2} and \sum_{k=1}^n k^2 = \frac{k(k+1)(2k+1)}{6}.
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