$\sum_{k=1}^{n} \binom{n}{k}k^{r}$
70248
Closed-form expression for $\sum_{k=0}^n\binom{n}kk^p$ for integers $n,\,p$
132249
Sum of every $k$th binomial coefficient.
49771
Prove that $\sum\limits_{k=0}^r\binom{n+k}k=\binom{n+r+1}r$ using combinatoric arguments.
119107
Is it true that $\sum_{k=3}^{n}\binom{n}{k}\binom{k}{3} = \binom{n}{3}\binom{n-3}{k-3}$?
130543
If $a_n = \sum\limits_{r=0}^n \binom{n}{r} b_r$, prove $(-1)^n b_n = \sum\limits_{s=0}^n \binom{n}{s} (-1)^s a_s$
107071
Prove $\sum_{k= 0}^{n} k \binom{n}{k} = n \cdot 2^{n - 1}$ using the binomial theorem
9522
Proving $k \binom{n}{k} = n \binom{n-1}{k-1}$
91239
Sum of binomial coefficients involving $n,p,q,r$
114414
Sum of $\sum_{k=1}^n\sin{k\theta}$
69498
What is $\sum_{r=0}^n \frac{(-1)^r}{\binom{n}{r}}$?
1258
Prove $3^n = \sum_{k=0}^n \binom {n} {k} 2^k$
116722
Simplifying $\big(\sum_{i=0}^n\binom{k}{i}\binom{M-k}{n-i}\frac{k-i}{M-n}\big)/{\binom{M}{n}}$
92440
Let $n \in \mathbb{Z}^+$, prove the identity $ \sum_{k=1}^{n-1} \binom {n} {k} \frac{kn^{n-k}}{k+1}=\frac{n(n^{n}-1)}{n+1}$
35342
Show that $ \sum_{n=2}^m \binom{n}{2} = \binom{m+1}{3}$
92704
Prove that $\sum_{k=1}^{n}k\binom{n}{k}=n\cdot2^{n-1}$
147910
Prove that $\sum_{n=0}^N\binom{2N-n}N2^n(n+1)=(1+2N)\binom{2N}N$
1139
Combinatorial proof of $\sum_{j=0}^{k} \binom{n}{j} = \sum_{j=0}^k \binom{n-1-j}{k-j}2^j$