Prove that $\sum_{r=0}^n({n\choose r})^2={2n\choose n }$
139426
Proving $ \binom n 0 ^2 + \binom n 1 ^2 + \dots + \binom n n ^2 = \binom { 2n} n $ without induction
146533
Having problems proving $ \sum_{r=0}^{n}(r+1) \binom{n}{r} = (n+2)2^{n-1} $
37650
Find $\sum_{m=0}^n\ (-1)^m m^n {n \choose m}$
123955
How to compute $\sum\limits_{k=0}^n (-1)^k{2n-k\choose k}$?
114911
Evaluate $\sum_{k = 0}^{n} {n\choose k} k^m$
48434
Prove that $\sum\limits_{k=0}^m\binom{m}k\binom{n}{r+k}=\binom{m+n}{m+r}$ using combinatoric arguments.
114031
Show that $\sum_{n=0}^\infty r^n e^{i n \theta} = \frac{1- r\cos(\theta)+i r \sin(\theta)}{1+r^2-2r\cos(\theta)}$
345
Proving $ \sum _{k=0} ^m \binom nk \binom{n-k}{m-k} = 2^m \binom {n}{m}$.
71534
Combinatorial explanation for why $n^2 = {n \choose 2} + {n+1 \choose 2}$
27922
prove that $\binom{n}{0}^2+\binom{n}{1}^2+\binom{n}{2}^2+\cdots+\binom{n}{n}^2=\binom{2n}{n}$
151757
Simplify $\sum_{i=0}^{N/2}{N-i\choose i}$
79124
Proving $\binom {n-1}{r-1}=\sum_{k=0}^r(-1)^k\binom r k \binom{n+r-k-1}{r-k-1}$
47670
Algebraic proof of $\sum_{i=0}^k{{n \choose i}{m \choose {k-i}}}= {{m+n}\choose k}$
60860
Combinatorial proof of $\sum_{i = 0}^{n} \binom{i}{r - 1} = \binom{n + 1}{r}$ and then use result to find a formula for $1^2 + 2^2 + \ldots + n^2$
118471
Prove that $1-\sum_{i=1}^n a_i < \prod_{i=1}^n(1-a_i) $
114173
Explanation of the formulas for sums $\sum nr^n$ and $\sum n^2 r^n$
5272
Proving that $\sum_{k=0}^{n}\frac{(-1)^k}{{n\choose k}}=[1+(-1)^n] \frac{n+1}{n+2}.$