Value of sum of binomials: $P = \binom{N}{0}-\binom{N}{1}+\binom{N}{2}-\binom{N}{3}+ \dotsb + (-1)^N\binom{N}{N}$
5049
Proving $\sum_{k=0}^n(-1)^k\binom nk=0$
44621
Simple binomial theorem proof: $\sum_{j=0}^{k} \binom{a+j}j = \binom{a+k+1}k$
36361
How to calculate the summation $(\sum_{p = k}^{n} \binom{n}{p}) / 2^n$ quickly?
71797
Combinatorial argument for $\sum\limits_{k=i}^{n}\binom{n}{k}\binom{k}{i} = \binom{n}{i}2^{n-i}$
114280
Computing $\sum_{n=1}^∞\frac{1}{(n+1)(n+2)(n+3)....(n+p)}$
1158
How to prove this binomial identity $\sum_{r=0}^n {r {n \choose r}} = n2^{n-1}$?
131400
Solution check: Let $f_n$ be fibonacci numbers. Prove: $\sum_{k=0}^{n-1} \binom{n+k}{2k+1} = f_{2n-1}$ and $\sum_{k=0}^n \binom{n+k}{2k} = f_{2n}$
126888
How can you show that $\binom {n}{7}=\sum_{k=7}^n \binom {k-1} {6}$?
74294
How to simplify $\sum_{i=1}^{k}\binom{n + i - 1}{i}$?
92704
Prove that $\sum_{k=1}^{n}k\binom{n}{k}=n\cdot2^{n-1}$
70248
Closed-form expression for $\sum_{k=0}^n\binom{n}kk^p$ for integers $n,\,p$
7083
Alternating sum of binomial coefficients: given $n \in \mathbb N$, prove $\sum^n_{k=0}(-1)^k {n \choose k} = 0$
146762
Closed form for $\sum_{k=0}^{n} \binom{n}{k}\frac{(-1)^k}{(k+1)^2}$
116722
Simplifying $\big(\sum_{i=0}^n\binom{k}{i}\binom{M-k}{n-i}\frac{k-i}{M-n}\big)/{\binom{M}{n}}$
27922
prove that $\binom{n}{0}^2+\binom{n}{1}^2+\binom{n}{2}^2+\cdots+\binom{n}{n}^2=\binom{2n}{n}$
51864
$\left( \frac{1 \cdot 2}{p} \right) + \left( \frac{2 \cdot 3}{p} \right) + \cdots + \left( \frac{(p-2)(p-1)}{p} \right) = -1$
79124
Proving $\binom {n-1}{r-1}=\sum_{k=0}^r(-1)^k\binom r k \binom{n+r-k-1}{r-k-1}$