MATH

Prove that
C(n+1,r) = C(n,r)+ C(n,r-1)

~~~~~
I know the defintion of a combination is... C= n!/(n-r)!r! but i still cant get left side to equal right side. help.

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asked by Alex
  1. http://answers.yahoo.com/question/index?qid=20071130201604AAy0UBp

    That should guide you. IT is not exactly the same, but you can follow it.

  2. defn.
    C(n,r) = n!/((n-r)!r!)

    LS = (n+1)!/(r!(n+1-r)!)

    RS = n!/((n-r)!r!) + n!/((r-1)!(n-r-1)!)

    now recall that something like 5x4! = 5!
    then in the same way if I have n! and multiply it by n+1 I would get (n+1)!

    so back to
    RS = n!/((n-r)!r!) + n!/((r-1)!(n-r-1)!)
    I will multiply the top and bottom of the first fraction by
    (n-r+1)
    and I will multiply the top and bottom of the second fraction by r
    simplifying this (without typing that mess) I get
    RS = n!(n-r+1)/((n-r=1)!r!) + (n!r/(n-r+1)!r!)
    notice we now have a common denominator, so we add up the top.
    But I see a common factor of n! in the top , so
    RS = n!(n-r+1 - r)/((n-r+1)!r!)
    = n!(n+1)/((n-r+1)!r!)
    = (n+1)!/((n-r+1)!r!)
    = LS

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    posted by Reiny

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