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Combinatoric Proof

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anemone

MHB POTW Director
Staff member
Feb 14, 2012
3,756
Show that \(\displaystyle {{n\choose 2}\choose 2 }=3 {n+1\choose 4}\)
 

kaliprasad

Well-known member
Mar 31, 2013
1,322
I would expect some one to solve combinatorically but I provide algebragically


writing (n k) for nCk

We have (n 2) = n(n-1)/2

So ((n 2) 2) = ((n(n-1)/2) (n(n-1)/2-1))/2
= n(n-1)/2 ( n^2-n -2)/ 4
= n(n-1)(n-2)(n+1)/8
= 3( n(n-1)(n-2)(n+1)/24
= 3 ( n+1)!/(4!(n+1-4)! Multiply numerator and dnominator by (n-3)!
= 3 (n + 1 4)