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Inequality involving two unknowns and factorials

  1. Jul 7, 2007 #1
    I need to prove the following:
    n!/r! >= r^(n-r)

    With r and n as natural numbers and n>=r

    I know the LHS will end up being (n-r) terms long as the first r! will cancel out of n! (n>=r), but as they're both unknown, I just left it as


    and I looked at the RHS as r^n/r^r, but I'm not sure how to work with these two sides. :uhh:

    Any tips on different ways to approach this would be greatly appreciated.


    Update: Is it enough to do the following, using induction:
    Prove that for n=1, it works as 1 >= 1
    and then, assuming the inequality holds for n, try n+1:
    (n+1)!/r! >= r^[(n+1)-r]
    which is just
    n!(n+1)/r! >= r^(n-r)*r
    and, as n>=r, (n+1)>=r, so, by induction, it's holds for all n.
    Last edited: Jul 8, 2007
  2. jcsd
  3. Jul 8, 2007 #2


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    That's the way I would do it! For a moment I thought- "He's assuming r= 1 and he can't do that", but you can. Since r is a natural number, less than or equal to n, if n= 1 then r= 1.
  4. Jul 8, 2007 #3
    LHS= (r+1)*(r+2)...*n , there are n-r terms and each term is greater than r . Hence, the LHS is greater than r^(n-r), right?
  5. Jul 8, 2007 #4

    Gib Z

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    Ahh I like the way this new guy thinks :) Much simpler than induction, original and just plain :) which is good lol.
  6. Jul 8, 2007 #5
    Thanks to everyone who responded!

    I appreciate the simplicity of your response, huyen_vyvy.

    One small note: it's a she who assumed r=1, not a he.
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