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- #1

- Feb 14, 2012

- 3,756

Show that \(\displaystyle {{n\choose 2}\choose 2 }=3 {n+1\choose 4}\)

- Thread starter anemone
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- Thread starter
- Admin
- #1

- Feb 14, 2012

- 3,756

Show that \(\displaystyle {{n\choose 2}\choose 2 }=3 {n+1\choose 4}\)

- Mar 31, 2013

- 1,322

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)