$\sum_{r=0}^{n-k} {n \choose r}{n \choose {r+k}}={n \choose {n-k}}$
87479
Closed form for $\sum_{k=0}^{n-p}\binom{n}{k}\binom{n}{p+k}$
103840
Calculate $\sum_{k=1}^n k2^{n-k}$
79124
Proving $\binom {n-1}{r-1}=\sum_{k=0}^r(-1)^k\binom r k \binom{n+r-k-1}{r-k-1}$
93813
How to show that $\sum\limits_{k=0}^n (-1)^k\tfrac{{ {n}\choose{k}}}{{ {x+k}\choose{k}}} = \frac{x}{x+n}$
119107
Is it true that $\sum_{k=3}^{n}\binom{n}{k}\binom{k}{3} = \binom{n}{3}\binom{n-3}{k-3}$?
35381
Sum with binomial coefficients: $\sum_{k=1}^m  \frac{1}{k}{m \choose k} $
24588
Proving the combinatorial identity ${n \choose k} = {n-2\choose k-2} + 2{n-2\choose k-1} + {n-2\choose k}$
959
Prove that ${{n}\choose{k}} = {{n - 2}\choose{k}} + 2{{n - 2}\choose{k - 1}} + {{n - 2}\choose{k - 2}}$.
1167
How to prove that $\sum_{k=0}^{n}(-1)^k \binom {n}{k}=0 $
68080
Evaluation of $\sum_{k=0}^n{n\choose k}^2u^k$
116491
Closed form for $\sum_{k=0}^{m} {\binom {m}{k}} a^{k} (b+ck)^N$
30472
Evaluate $\sum_{k=0}^{n} {n \choose k}{m \choose k}$ for a given $n$ and $m$.
28027
Find $\sum_{k=0}^{n}4^k \binom{n}{k}$
60031
Showing $\sum\limits_{j=0}^M \frac{M \choose j}{N+M \choose j} = \frac{N+M+1}{N+1}$
148949
Find the value of $\sum_{k=1}^{n} k \binom {n} {k}$
142534
Proof of $k {n\choose k} = n {n-1 \choose k-1}$ using direct proof
91239
Sum of binomial coefficients involving $n,p,q,r$