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Thread: Using Recursion Equations

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

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