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Proving a perfect square with factorials

  1. Sep 14, 2005 #1
    The tex seems to be showing different problems than the ones I'm typing... maybe it's just me, but if what I'm talking about doesn't seem to make any sense, please quote my message to see what I've actually typed in the tex tags.

    If n is a positive integer and n > 1, prove that nC2 + (n-1)C2 is a perfect square.

    Now, expanded, this is what they look like...

    [TEX]\frac{n!}{2!(n-2)!} + \frac{(n-1)!}{2!(n-3)!}[/TEX]

    I'm not sure whether to multiply by the denominators or whether I'm simply supposed to try to simplify each one on its own, or a combination of the two. Moreover, I'm certain I'll still have at least one factorial left when I'm through, and how can I prove that any function with a factorial in it is a perfect square? Could someone give me a nudge in the right direction?
  2. jcsd
  3. Sep 14, 2005 #2


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    You should be able to pull a few terms out of the top so that you can cancel the factorial on the bottom.
  4. Sep 14, 2005 #3
    Do you mean expanding the n! on the top with both functions until the (n-2)! on the left one and the (n-3)! on the right one cancel out the one at the top?
  5. Sep 14, 2005 #4


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    That should work, but I think it's a little easier if you add the fractions first. I'm sorry, I should have been clearer. I was assuming you were stuck at the step you mentioned where you had one term with factorials in it, and I was going from there.
    Last edited: Sep 14, 2005
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