Proving $\sum_{k=0}^n k^2{n \choose k} = n(n+1)2^{n-2}$
16347
Induction: $\sum_{k=0}^n \binom nk k^2 = n(1+n)2^{n-2}$
36436
Proving $\sum_{k=1}^n k k!=(n+1)!-1$
87320
Proving identities like $\sum_{k=1}^nk{n\choose k}^2=n{2n-1\choose n}$ combinatorially
34299
Combinatorial proof of $\sum_{k=0}^n k \cdot k! = (n+1)! -1$
345
Proving $ \sum _{k=0} ^m \binom nk \binom{n-k}{m-k} = 2^m \binom {n}{m}$.
147910
Prove that $\sum_{n=0}^N\binom{2N-n}N2^n(n+1)=(1+2N)\binom{2N}N$
116461
Prove the identity $\sum_{k=0}^{n}(-1)^{k}\binom{n}{k}=0$ using a combinatorial interpretation of the positive and negative terms
7083
Alternating sum of binomial coefficients: given $n \in \mathbb N$, prove $\sum^n_{k=0}(-1)^k {n \choose k} = 0$
113321
Hint proving this $\sum_{k=0}^{n}\binom{2n}{k}k=n2^{2n-1}$
11612
How to show that $\sum_{k=1}^n k(n+1-k)=\binom{n+2}3$?
1125
Inductive proof that ${2n\choose n}=\sum{n\choose i}^2.$
14197
show that $\sum_{i=0}^l \dfrac{{l\choose i}}{{n\choose k + i}} = \dfrac{n+1}{(n-l+1) {n-l\choose k}}$
959
Prove that ${{n}\choose{k}} = {{n - 2}\choose{k}} + 2{{n - 2}\choose{k - 1}} + {{n - 2}\choose{k - 2}}$.
24588
Proving the combinatorial identity ${n \choose k} = {n-2\choose k-2} + 2{n-2\choose k-1} + {n-2\choose k}$
1167
How to prove that $\sum_{k=0}^{n}(-1)^k \binom {n}{k}=0 $
140877
Evaluate the sum $\sum_{k=1}^n {n \choose k-1} {n \choose k}$
153226
Proof of $\sum_{k = 1}^{n} \frac{1}{k^{2}} < 2 - \frac{1}{n}$