$0^2{{n}\choose{0}} + 1^2{{n}\choose{1}} + 2^2 {{n}\choose{2}} + ... +n^2{{n}\choose{n}} = n (1+n)* 2^{n-2}, n \in \mathbb{N} \setminus \{0\}$ proof.
16347
Induction: $\sum_{k=0}^n \binom nk k^2 = n(1+n)2^{n-2}$
70827
Prove that $\sum_{i = n - b}^{a - m} {a - i \choose m}{b + i \choose n} = {a + b + 1 \choose m + n + 1}$
96894
Prove $f_{n+1}^2+f_n^2=f_{2n+1}$
44161
Using induction to prove $\sum_{k=0}^{N-1} 2^k = 2^N-1$
54442
Sum of $k {n \choose k}$ is $n2^{n-1}$
71014
Prove that 2(nC0+nC2...nC$\frac{(N-1)}{2}$=$2^n$
92440
Let $n \in \mathbb{Z}^+$, prove the identity $ \sum_{k=1}^{n-1} \binom {n} {k} \frac{kn^{n-k}}{k+1}=\frac{n(n^{n}-1)}{n+1}$
60860
Combinatorial proof of $\sum_{i = 0}^{n} \binom{i}{r - 1} = \binom{n + 1}{r}$ and then use result to find a formula for $1^2 + 2^2 + \ldots + n^2$
16177
Proof by Induction: $\sum_0^nx^i=(1-x^{n+1})/(1-x)$
1139
Combinatorial proof of $\sum_{j=0}^{k} \binom{n}{j} = \sum_{j=0}^k \binom{n-1-j}{k-j}2^j$
44484
Combinatorial identity $i{n \choose i } =n {n-1 \choose i - 1}$
5272
Proving that $\sum_{k=0}^{n}\frac{(-1)^k}{{n\choose k}}=[1+(-1)^n] \frac{n+1}{n+2}.$
6366
Combinatorial proof of summation of $\sum\limits_{k = 0}^n {n \choose k}^2= {2n \choose n}$
7083
Alternating sum of binomial coefficients: given $n \in \mathbb N$, prove $\sum^n_{k=0}(-1)^k {n \choose k} = 0$
52206
How to prove with double counting technique that $1+2+\dots+2^n=2^{n+1} -1$?
72026
Induction proof of $\sum_{k=1}^{n} \binom n k = 2^n -1 $
116461
Prove the identity $\sum_{k=0}^{n}(-1)^{k}\binom{n}{k}=0$ using a combinatorial interpretation of the positive and negative terms