Show that $\sum_{k=1}^n k^2 {n\choose k} = n(n+1)2^{n-2}$.
16347
Induction: $\sum_{k=0}^n \binom nk k^2 = n(1+n)2^{n-2}$
133905
Intuitive proof of $\sum_{k=1}^{n} \binom{n}{k} k^{k-1} (n-k)^{n-k} = n^n$
32791
Proving this binomial identity $\sum_{k=0}^n {n+k \choose k} \frac{1}{2^{k}}= 2^{n}$
25826
Proof of a combinatorial identity: $\sum\limits_{i=0}^n {2i \choose i}{2(n-i)\choose n-i} = 4^n$
70282
Help finding a combinatorial proof of $k {n \choose k } = n {n - 1 \choose k -1}$
58428
Proving $\sum_{k=0}^{n} {k \choose a} {n-k \choose b} = {n+1\choose a+b+1}$
30672
Show that $\sum\limits_{k=0}^n\binom{2n}{2k}^{\!2}-\sum\limits_{k=0}^{n-1}\binom{2n}{2k+1}^{\!2}=(-1)^n\binom{2n}{n}$
5272
Proving that $\sum_{k=0}^{n}\frac{(-1)^k}{{n\choose k}}=[1+(-1)^n] \frac{n+1}{n+2}.$
84213
How to show $n\sum_{i=1}^n {x_i^2} \ge (\sum_{i=1}^n{x_i})^2$
113321
Hint proving this $\sum_{k=0}^{n}\binom{2n}{k}k=n2^{2n-1}$
68080
Evaluation of $\sum_{k=0}^n{n\choose k}^2u^k$
58247
Prove $\sum_{k=0}^n \binom{n+k}{k}=\binom{2n+1}{n}$
52113
Proof by Induction $\sum_{i=0}^{k-1} 2^i = 2^k-1$
26901
induction proof: $\sum_{k=1}^nk^2 = \frac{n(n+1)(2n+1)}{6}$
111098
Prove $2^{2n} = \sum_{k=0}^{n}\binom{2n+1}{k}$
2455
Prove that $\left(\sum^n_{k=1}x_k\right)\left(\sum^n_{k=1}y_k\right)\geq n^2$
96903
How to show via inductive proof (and without the use of the binomial theorem) that $ {n\choose k} \in \mathbb{N} $?