$\displaystyle\sum_{k\geq 0}^{} \binom{n}{2k} = \sum_{k \geq 0}^{} \binom{n}{2k+1}$
21152
Exactly half of the elements of $\mathcal{P}(A)$ are odd-sized
11612
How to show that $\sum_{k=1}^n k(n+1-k)=\binom{n+2}3$?
119107
Is it true that $\sum_{k=3}^{n}\binom{n}{k}\binom{k}{3} = \binom{n}{3}\binom{n-3}{k-3}$?
48200
Simplify $\sum_{i=0}^n (i+1)\binom ni$
345
Proving $ \sum _{k=0} ^m \binom nk \binom{n-k}{m-k} = 2^m \binom {n}{m}$.
1167
How to prove that $\sum_{k=0}^{n}(-1)^k \binom {n}{k}=0 $
147910
Prove that $\sum_{n=0}^N\binom{2N-n}N2^n(n+1)=(1+2N)\binom{2N}N$
26446
Find the sum : $\displaystyle\sum_{i=0}^n \frac{2^i}{1+x^{2^{i}}}$
131326
How to find $\sum_{r\ge 0} \binom{n}{r}\binom{n-r}{r} 2^{n-2r}$?
30338
How to prove that $\sum_{k=0}^n \binom nk k^2=2^{n-2}(n^2+n)$
71797
Combinatorial argument for $\sum\limits_{k=i}^{n}\binom{n}{k}\binom{k}{i} = \binom{n}{i}2^{n-i}$
87589
Evaluate $\binom{n}{0}+\binom{n}{2}+\binom{n}{4}+...+\binom{n}{2k}$
142994
How to show $\sum_{i=1}^{n} \binom{i}{2}=\binom{n+1}{3}$?
19571
Combinatorial proof for $\sum_{k = 0}^n \binom {r+k} k = \binom {r + n + 1} n$
155005
How to show $\sum_{k=n}^\infty{\frac{1}{k!}} \leq \frac{2}{n!}$
104704
Evaluate $\sum\limits_{k=0}^n \binom{n}{k}$ combinatorially
143047
find $\sum_{k=0}^{t}(-1)^k\binom{t}{k}^2$ for odd t then for even t