Proof of $S_{n,r} = \frac{1}{r!} \sum_{k=0}^r (-1)^{r-k} \binom{r}{k} k^n$ with inclusion-exclusion principle
133070
Finding a closed form expression for the recurrence relation $a(n,k+1)=(k+1)^n+\sum_{m=1}^k {k+1\choose m}a(n,m)$
35956
Proof that a sum of Bernoulli rvs has Binomial distribution
18046
Prove the following equality: $\sum_{k=0}^n\binom {n-k }{k} = F_n$
1167
How to prove that $\sum_{k=0}^{n}(-1)^k \binom {n}{k}=0 $
117775
Combinatorial proof of a Stirling number identity.
66019
Revisited: Binomial Theorem: An Inductive Proof
67650
Short and intuitive proof that $\left(\frac{n}{k}\right)^k \leq \binom{n}{k}$
63510
Prove: $\binom{n}{k}^{-1}=(n+1)\int_{0}^{1}x^{k}(1-x)^{n-k}dx$ for $0 \leq k \leq n$
145048
Prove $\sum_{k = 0}^{n}(-1)^{n - k} \binom{n}{k} \cdot k^n = n!$ and $\sum_{k = 0}^{n}(-1)^{n - k} \binom{n}{k} \cdot k^m = 0$
29828
Combinatorial proof of $\sum^{n}_{i=1}\binom{n}{i}i=n2^{n-1}$.
27585
Combinatorial proof of Stirling Number Identity (First Kind)
84010
Combinatorial proof that $\sum \limits_{k=0}^n \binom{2k}{k} \binom{2n-2k}{n-k} (-1)^k = 2^n \binom{n}{n/2}$ when $n$ is even
69498
What is $\sum_{r=0}^n \frac{(-1)^r}{\binom{n}{r}}$?
59252
Show $\sum_{k=0}^{m} \binom{n-k}{m-k}=\binom{n+1}{m}$
131326
How to find $\sum_{r\ge 0} \binom{n}{r}\binom{n-r}{r} 2^{n-2r}$?
58247
Prove $\sum_{k=0}^n \binom{n+k}{k}=\binom{2n+1}{n}$
153226
Proof of $\sum_{k = 1}^{n} \frac{1}{k^{2}} < 2 - \frac{1}{n}$