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Math Help - Koshy-Elementary Number Theory with Applications...Induction, sums and binomial.

  1. #1
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    Koshy-Elementary Number Theory with Applications...Induction, sums and binomial.

    Well, I have a problem. These are some exercises from the Koshy's book about Number Theory...

    The first is:
    Verify that $\tbinom{n}{r}=\frac{n}{r}\tbinom{n-1}{r-1}$.

    That is easy, I did this:
    Koshy-Elementary Number Theory with Applications...Induction, sums and binomial.-ncombr.png
    Then the next one was:
    Show that:
    Koshy-Elementary Number Theory with Applications...Induction, sums and binomial.-sumatoriacomb.png

    I tried to prove it using Mathematical Induction, but I don't get to the result...
    I'll show what I've done:
    Koshy-Elementary Number Theory with Applications...Induction, sums and binomial.-inductionproof.png
    But I don't know how to get to where I wanna get...
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  2. #2
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    Re: Koshy-Elementary Number Theory with Applications...Induction, sums and binomial.

    Hi,
    You didn't say you must have a proof by induction (I don't see an easy inductive proof). Here's an easy proof:

    Koshy-Elementary Number Theory with Applications...Induction, sums and binomial.-mhfbinomialsums.png

    The formula represents two different way of counting the ways one can choose n people from 2n people with exactly n men and n women -- a term of the sum represents choosing r men and thus n-r women; letting r vary from 0 to n we get the total number of ways of choosing the n people. This is in fact the way I remember the formula.

    D. E. Knuth's Art of Computer Programming, Vol I has a wealth of combinatoric formulas like this. Perhaps even more can be found in Concrete Mathematics by Knuth, et al.
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