1. ## Combination

Prove that (n+1)C4=((nc2)c2)/3 ,where n>=4

2. ## Re: Combination

$\displaystyle \binom{n}{2} = \frac{n(n-1)}{2} = \frac{n^2 - n}{2}$. Also, $\displaystyle \binom{\binom{n}{2}}{2} = \frac{\frac{n^2 - n}{2} \frac{n^2 - n - 2}{2}}{2} = \frac{(n^2 - n)(n^2 - n - 2)}{8}$. So you can simplify the RHS. Then use algebra.

3. ## Re: Combination

Hello, Swarnav!

$\displaystyle \text{Prove: }\:{n+1\choose4} \:=\:\frac{1}{3}{{n\choose2}\choose2}\;\text{where }n \ge 4$

$\displaystyle \text{RHS: }\:\frac{1}{3}\cdot{\frac{n(n-1)}{2\cdot1} \choose 2} \;=\;\frac{1}{3}\cdot\frac{\left(\frac{n(n-1)}{2}\right)\left(\frac{n(n-1)}{2} - 1\right)}{2\cdot 1} \;=\;\frac{1}{3}\cdot\frac{\left(\frac{n(n-1)}{2}\right)\left(\frac{n(n-1)-2}{2}\right)}{2\cdot 1}$

. . . $\displaystyle =\;\frac{1}{3}\cdot\frac{n(n-1)(n^2-n-2)}{4\cdot2\cdot1} \;=\; \frac{1}{3}\cdot\frac{n(n-1)(n+1)(n-2)}{4\cdot2\cdot1}$

. . . $\displaystyle =\;\frac{(n+1)n(n-1)(n-2)}{4\cdot3\cdot2\cdot1} \;=\;{n+1\choose4} \;=\;\text{LHS}$