Proving $\sum_{k=0}^n{2k\choose k}{2n-2k\choose n-k}=4^n$
7143
Identity for convolution of central binomial coefficients: $\sum\limits_{k=0}^n \binom{2k}{k}\binom{2(n-k)}{n-k}=2^{2n}$
148442
Show that the sum $\sum_{k = 0}^n 2^k \binom{n}{k}$ is equal to $3^n$
10917
Give a combinatorial proof: $ n(n+1)2^{n-2} = \sum_{k=1}^{n}k^2\binom{n}{k} $
113321
Hint proving this $\sum_{k=0}^{n}\binom{2n}{k}k=n2^{2n-1}$
24588
Proving the combinatorial identity ${n \choose k} = {n-2\choose k-2} + 2{n-2\choose k-1} + {n-2\choose k}$
84010
Combinatorial proof that $\sum \limits_{k=0}^n \binom{2k}{k} \binom{2n-2k}{n-k} (-1)^k = 2^n \binom{n}{n/2}$ when $n$ is even
30338
How to prove that $\sum_{k=0}^n \binom nk k^2=2^{n-2}(n^2+n)$
17638
Sum with binomial coefficients: $\sum_{k=0}^{n}{2n\choose 2k}$
1167
How to prove that $\sum_{k=0}^{n}(-1)^k \binom {n}{k}=0 $
82742
Proving that ${x +y+n- 1 \choose n}= \sum_{k=0}^n{x+n-k-1 \choose n-k}{y+k-1 \choose k} $
52113
Proof by Induction $\sum_{i=0}^{k-1} 2^i = 2^k-1$
345
Proving $ \sum _{k=0} ^m \binom nk \binom{n-k}{m-k} = 2^m \binom {n}{m}$.
50226
Prove the identity $\sum_{k=0}^{n}\sum_{r=0}^{k} \binom{k}{r} \binom{n}{k} = 3^n$
70128
Prove that $(fg)^{(n)} = \sum_{k=0}^n \binom{n}{k}f^{(k)}(x)g^{(n-k)}(x)$
63879
Prove that $\sum_{k=0}^r {m \choose k} {n \choose r-k} = {m+n \choose r}$
47670
Algebraic proof of $\sum_{i=0}^k{{n \choose i}{m \choose {k-i}}}= {{m+n}\choose k}$
959
Prove that ${{n}\choose{k}} = {{n - 2}\choose{k}} + 2{{n - 2}\choose{k - 1}} + {{n - 2}\choose{k - 2}}$.