Prove $2 \cdot \sum\limits_{k = 0}^{n} \binom{2n}{2k} = 2^{2n}$ without induction
17638
Sum with binomial coefficients: $\sum_{k=0}^{n}{2n\choose 2k}$
111951
Prove $\binom{n}{m}\binom{m}{k} = \binom{n}{k}\binom{n-k}{m-k}$
4991
Proving by induction that $ \sum_{k=0}^n{n \choose k} = 2^n$
112374
Evaluate $ \binom{n}{0}+\binom{n}{2}+\binom{n}{4}+\cdots+\binom{n}{2k}+\cdots$
122006
Proving combinatorial identity $\sum_{k=0}^{n}(-1)^k \binom{2n-k}k 2^{2n-2k} = 2n+1$ "directly"
30338
How to prove that $\sum_{k=0}^n \binom nk k^2=2^{n-2}(n^2+n)$
108401
Sum of binomial coefficients $\sum_{k=0}^{n}(-1)^k\binom{n}{k}\binom{2n - 2k}{n - 1} = 0$
23557
Algebraic Proof that $\sum\limits_{i=0}^n \binom{n}{i}=2^n$
36786
Prove $\sum\limits_{i=0}^n\binom{i+k-1}{k-1}=\binom{n+k}{k}$ (a.k.a. Hockey-Stick Identity)
126001
Prove by Double Counting Method $\sum\limits_{k = 0}^m \binom{m}{k}\binom{n}{r + k} = \binom{m + n}{m + r}$
113321
Hint proving this $\sum_{k=0}^{n}\binom{2n}{k}k=n2^{2n-1}$
154722
How do I prove $2 \binom n2 + 4 \binom n4 + 6 \binom n6 + \cdots = n 2^{n-2}$
143113
Non-inductive, not combinatorial proof of $\sum_{i \mathop = 0}^n \binom n i^2 = \binom {2 n} n$
7143
Identity for convolution of central binomial coefficients: $\sum\limits_{k=0}^n \binom{2k}{k}\binom{2(n-k)}{n-k}=2^{2n}$
89576
Proof of a binomial identity $\sum_{k=0}^n {n \choose k}^{\!2} = {2n \choose n}.$
118571
Why is $\sum\limits_{k=0}^{n}(-1)^k\binom{n}{k}^2=(-1)^{n/2}\binom{n}{n/2}$ if $n$ is even?
59252
Show $\sum_{k=0}^{m} \binom{n-k}{m-k}=\binom{n+1}{m}$