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Math Help - A store has n different products for sale

  1. #1
    MHF Contributor alexmahone's Avatar
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    A store has n different products for sale

    A store has n different products for sale. Each of them has a different price that is at least one dollar, at most n dollars, and is a whole dollar. A customer only has the time to inspect k different products. After doing so, she buys the product that has the lowest price among the k products she inspected. Prove that on average, she will pay \frac{n+1}{k+1} dollars.

    My attempt:

    There are ^nC_k ways to select k products out of n. The maximum amount she will ever pay is n-k+1.

    There are ^{n-1}C_{k-1} ways in which she ends up paying 1 dollar.
    There are ^{n-2}C_{k-1} ways in which she ends up paying 2 dollars.
    ... and so on.

    Average = \frac{1*^{n-1}C_{k-1}+2*^{n-2}C_{k-1}+...+(n-k+1)*^{k-1}C_{k-1}}{^nC_k}

    How do I proceed?
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  2. #2
    Member Traveller's Avatar
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    Re: A store has n different products for sale

    Try to use Pascal's identity.
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