Show $\sum_{ r = 0}^{n} {(\binom n r)}^{2} = \frac{(2n)!}{(n!)^{2}}$
6366
Combinatorial proof of summation of $\sum\limits_{k = 0}^n {n \choose k}^2= {2n \choose n}$
66568
Number Theory : Show that $\sum_{i=0}^\infty$ $ [\frac{n}{2^i}+\frac{1}{2}]$ $=$ $2n$
147910
Prove that $\sum_{n=0}^N\binom{2N-n}N2^n(n+1)=(1+2N)\binom{2N}N$
57418
How to prove this statement: $\binom{r}{r}+\binom{r+1}{r}+\cdots+\binom{n}{r}=\binom{n+1}{r+1}$
81729
Combinatorial proof of $\sum_{k=0}^{n}(-1)^{k}\binom{n}{k}(l-k)^n=n!$, using inclusion-exclusion
155769
Prove $\sum_{k= 0}^{n} k \binom{n}{k} = n \cdot 2^{n-1}$ using the binomial theorem for $n\geq1$
9046
Prove that $\sum_{k=0}^{m}\frac{\binom{m}{k}}{\binom{n}{k}}=\frac{n+1}{n+1-m}$
97684
Show $\frac{(2n)!}{n!\cdot 2^n}$ is an integer for $n$ greater than or equal to $0$
76319
How to prove $\sum\limits_{r=0}^n \frac{(-1)^r}{r+1}\binom{n}{r} = \frac1{n+1}$?
57549
Finding the sum $\sum_{k=1}^rk^2\binom {n-k}{r-k}$
46840
How to prove that $\sum_{r=0}^n\binom{n}{r}2^r=3^n$
72639
Show that $\sum_{k=0}^n\binom{3n}{3k}=\frac{8^n+2(-1)^n}{3}$
58247
Prove $\sum_{k=0}^n \binom{n+k}{k}=\binom{2n+1}{n}$
1139
Combinatorial proof of $\sum_{j=0}^{k} \binom{n}{j} = \sum_{j=0}^k \binom{n-1-j}{k-j}2^j$
111098
Prove $2^{2n} = \sum_{k=0}^{n}\binom{2n+1}{k}$
148949
Find the value of $\sum_{k=1}^{n} k \binom {n} {k}$
44111
Find the value of $\sum_{k=1}^{n}k\binom{n}{k}$?