Evaluate $ \binom{n}{0}+\binom{n}{2}+\binom{n}{4}+\cdots+\binom{n}{2k}+\cdots$
5049
Proving $\sum_{k=0}^n(-1)^k\binom nk=0$
75367
Find a simple formula for $\binom{n}{0}\binom{n}{1}+\binom{n}{1}\binom{n}{2}+...+\binom{n}{n-1}\binom{n}{n}$
117761
When is $\binom{n}{k}$ divisible by $n$?
59252
Show $\sum_{k=0}^{m} \binom{n-k}{m-k}=\binom{n+1}{m}$
72639
Show that $\sum_{k=0}^n\binom{3n}{3k}=\frac{8^n+2(-1)^n}{3}$
8806
Summation of binomial coefficients
143365
Is $\sum_{i=1}^{n-1}i=\binom{n}{2}$?
35342
Show that $ \sum_{n=2}^m \binom{n}{2} = \binom{m+1}{3}$
14986
Proving a combinatorics equality: $\binom{r}{r} + \binom{r+1}{r} + \cdots + \binom{n}{r} = \binom{n+1}{r+1}$
38705
Evaluating $\sqrt{1 + \sqrt{2 + \sqrt{4 + \sqrt{8 + \ldots}}}}$
14399
Calculate $\sum_{k=1}^n (-1)^{k+1} \binom{n}{k}\frac{1}{k}$
117200
Evaluation of $ \sum_{k=0}^n \cos k\theta $
71797
Combinatorial argument for $\sum\limits_{k=i}^{n}\binom{n}{k}\binom{k}{i} = \binom{n}{i}2^{n-i}$
104704
Evaluate $\sum\limits_{k=0}^n \binom{n}{k}$ combinatorially
70248
Closed-form expression for $\sum_{k=0}^n\binom{n}kk^p$ for integers $n,\,p$
74294
How to simplify $\sum_{i=1}^{k}\binom{n + i - 1}{i}$?
96945
General formula of $\sum_{n\ge1}{\binom{2n}{n}^{-k}}$.