Proving that $\sum\limits_{k=0}^{n} {{m+k} \choose{m}} = { m+n+1 \choose m+1 }$
36786
Prove $\sum\limits_{i=0}^n\binom{i+k-1}{k-1}=\binom{n+k}{k}$ (a.k.a. Hockey-Stick Identity)
2885
Prove that $\sum\limits_{i=1}^n 2i\binom{2n}{n-i}= n\binom{2n}{n}$
87320
Proving identities like $\sum_{k=1}^nk{n\choose k}^2=n{2n-1\choose n}$ combinatorially
49771
Prove that $\sum\limits_{k=0}^r\binom{n+k}k=\binom{n+r+1}r$ using combinatoric arguments.
35381
Sum with binomial coefficients: $\sum_{k=1}^m  \frac{1}{k}{m \choose k} $
40784
$\sum_{k=m}^{n}(-1)^k\binom{n}{k}\binom{k}{m}=0, n>m\geq 0$
86497
Prove that any rational can be expressed in the form $\sum\limits_{k=1}^n{\frac{1}{a_k}}$, $a_k\in\mathbb N^*$
126001
Prove by Double Counting Method $\sum\limits_{k = 0}^m \binom{m}{k}\binom{n}{r + k} = \binom{m + n}{m + r}$
89552
proving combinatorics identity - $\sum_{k=0}^m{n-k \choose m-k}={n+1 \choose m}$
114911
Evaluate $\sum_{k = 0}^{n} {n\choose k} k^m$
17375
When is the limit of a sum equal to the sum of limits?
157539
Prove: $\lim\limits_{n\to\infty}\frac{1}{n}\sum\limits_{k=1}^{n}\sqrt{1+\frac{k}{n}}=\frac{2}{3}(2\sqrt{2}-1)$
959
Prove that ${{n}\choose{k}} = {{n - 2}\choose{k}} + 2{{n - 2}\choose{k - 1}} + {{n - 2}\choose{k - 2}}$.
43885
Proving that $\sum\limits_{n = 0}^{2013} a_n z^n \neq 0$ if $a_0 > a_1 > \dots > a_{2013} > 0$ and $|z| \leq 1$
102531
Prove $\sum\limits^m_{k=0} \frac{2n-k\choose k}{2n-k\choose n}\frac{2n-4k+1}{2n-2k+1}2^{n-2k}=\frac{n\choose m}{2n-2m\choose n-m}2^{n-2m}$ for-
133181
Hint on how to prove the identity $\sum_{0 \leq k \leq n } \binom{k}{m} = \binom{n + 1}{m + 1}$
104704
Evaluate $\sum\limits_{k=0}^n \binom{n}{k}$ combinatorially