Show that $\sum_{k=0}^n \binom{m-k}{n-k} = \binom{m+1}{n}$
89552
proving combinatorics identity - $\sum_{k=0}^m{n-k \choose m-k}={n+1 \choose m}$
44621
Simple binomial theorem proof: $\sum_{j=0}^{k} \binom{a+j}j = \binom{a+k+1}k$
129304
Alternative combinatorial proof for $\sum\limits_{r=0}^n\binom{n}{r}\binom{m+r}{n}=\sum\limits_{r=0}^n\binom{n}{r}\binom{m}{r}2^r$
10917
Give a combinatorial proof: $ n(n+1)2^{n-2} = \sum_{k=1}^{n}k^2\binom{n}{k} $
18046
Prove the following equality: $\sum_{k=0}^n\binom {n-k }{k} = F_n$
19581
$\sum_{k=0}^n (-1)^k \binom{n}{k}^2$ and $\sum_{k=0}^n k \binom{n}{k}^2$
34299
Combinatorial proof of $\sum_{k=0}^n k \cdot k! = (n+1)! -1$
7083
Alternating sum of binomial coefficients: given $n \in \mathbb N$, prove $\sum^n_{k=0}(-1)^k {n \choose k} = 0$
49066
Give the combinatorial proof of the identity $\sum\limits_{i=0}^{n} \binom{k-1+i}{k-1} = \binom{n+k}{k}$
345
Proving $ \sum _{k=0} ^m \binom nk \binom{n-k}{m-k} = 2^m \binom {n}{m}$.
19571
Combinatorial proof for $\sum_{k = 0}^n \binom {r+k} k = \binom {r + n + 1} n$
145048
Prove $\sum_{k = 0}^{n}(-1)^{n - k} \binom{n}{k} \cdot k^n = n!$ and $\sum_{k = 0}^{n}(-1)^{n - k} \binom{n}{k} \cdot k^m = 0$
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
130223
Proving a special case of the binomial theorem: $\sum^{k}_{m=0}\binom{k}{m} = 2^k$
35342
Show that $ \sum_{n=2}^m \binom{n}{2} = \binom{m+1}{3}$
47670
Algebraic proof of $\sum_{i=0}^k{{n \choose i}{m \choose {k-i}}}= {{m+n}\choose k}$
142994
How to show $\sum_{i=1}^{n} \binom{i}{2}=\binom{n+1}{3}$?