$\sum _{k = 0}^ n \binom ka \binom {n-k}b = \binom {n+1}{a+b+1}$
58428
Proving $\sum_{k=0}^{n} {k \choose a} {n-k \choose b} = {n+1\choose a+b+1}$
83400
Sum of products of binomial coefficients: $ \sum_{l=0}^{n-j} \binom{M-1+l}{l} \binom{n-M-l}{n-j-l} = \binom{n}{j} $
14197
show that $\sum_{i=0}^l \dfrac{{l\choose i}}{{n\choose k + i}} = \dfrac{n+1}{(n-l+1) {n-l\choose k}}$
92704
Prove that $\sum_{k=1}^{n}k\binom{n}{k}=n\cdot2^{n-1}$
32791
Proving this binomial identity $\sum_{k=0}^n {n+k \choose k} \frac{1}{2^{k}}= 2^{n}$
126888
How can you show that $\binom {n}{7}=\sum_{k=7}^n \binom {k-1} {6}$?
146762
Closed form for $\sum_{k=0}^{n} \binom{n}{k}\frac{(-1)^k}{(k+1)^2}$
64803
Counting two ways, $\sum \binom{n}{k} \binom{m}{n-k} = \binom{n+m}{n}$
63510
Prove: $\binom{n}{k}^{-1}=(n+1)\int_{0}^{1}x^{k}(1-x)^{n-k}dx$ for $0 \leq k \leq n$
125712
Working with binomial coefficient $\sum_{k=0}^n (-1)^k \binom nk=0$
28027
Find $\sum_{k=0}^{n}4^k \binom{n}{k}$
130543
If $a_n = \sum\limits_{r=0}^n \binom{n}{r} b_r$, prove $(-1)^n b_n = \sum\limits_{s=0}^n \binom{n}{s} (-1)^s a_s$
7083
Alternating sum of binomial coefficients: given $n \in \mathbb N$, prove $\sum^n_{k=0}(-1)^k {n \choose k} = 0$
71797
Combinatorial argument for $\sum\limits_{k=i}^{n}\binom{n}{k}\binom{k}{i} = \binom{n}{i}2^{n-i}$
107071
Prove $\sum_{k= 0}^{n} k \binom{n}{k} = n \cdot 2^{n - 1}$ using the binomial theorem
14239
How to show $\sum_{k=0}^{n}\binom{n+k}{k}\frac{1}{2^k}=2^{n}$
9046
Prove that $\sum_{k=0}^{m}\frac{\binom{m}{k}}{\binom{n}{k}}=\frac{n+1}{n+1-m}$