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Math Help - Using Recursion Equations

  1. #1
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    Sep 2009
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    Using Recursion Equations

    Can someone help me on this problem?

    How many vectors (x1,x2......xk) are there for which each xi is a positive
    integer such that 1<= xi<=n and x1<x2<.......<xk?


    thanks in advance!
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  2. #2
    Super Member
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    Israel
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    If k=n, the answer is 1: (1,2,...,n)
    If k>n, the answer is 0.
    If k<n, we're actually looking at how many k-lengthed increasing sequences there are with entries in \{1,2,...,n\}

    You should note that for any k elements that we choose, there may only be one such sequence. Also, you can choose any k elements and then simply set them in order. This gives us that there are \binom{n}{k} options.
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