If $X_1,...,X_n$ are Bernoulli(p) iid then $X_1+...+X_n$ is Binomial(p,n)
35956
Proof that a sum of Bernoulli rvs has Binomial distribution
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
58247
Prove $\sum_{k=0}^n \binom{n+k}{k}=\binom{2n+1}{n}$
125220
Solving binomial theorem via induction
19571
Combinatorial proof for $\sum_{k = 0}^n \binom {r+k} k = \binom {r + n + 1} n$
35342
Show that $ \sum_{n=2}^m \binom{n}{2} = \binom{m+1}{3}$
2885
Prove that $\sum\limits_{i=1}^n 2i\binom{2n}{n-i}= n\binom{2n}{n}$
51587
Sum of Bernoulli random variables with different success probabilities
52557
Derivation of binomial coefficient in binomial theorem.
8806
Summation of binomial coefficients
4931
variation of the binomial theorem
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}$
40420
Proof by induction of Bernoulli's inequality: $(1 + x)^n \geq 1 + nx$
66996
Proof by induction of Bernoulli's inequality $ (1+x)^n \ge 1+nx$
142994
How to show $\sum_{i=1}^{n} \binom{i}{2}=\binom{n+1}{3}$?
7083
Alternating sum of binomial coefficients: given $n \in \mathbb N$, prove $\sum^n_{k=0}(-1)^k {n \choose k} = 0$
71564
Combinatorial proof of identity involving central binomial coefficients