Prove: ${n\choose 0}{m\choose k}+{n\choose 1}{m\choose {k-1}}+...+{n\choose k}{m\choose 0}={{m+n}\choose k}$
42763
How to prove Vandermonde's Identity: $\sum_{k=0}^{n}\binom{R}{k}\binom{M}{n-k}=\binom{R+M}{n}$?
121279
Proving an Combination formula $ \binom{n}{k} = \binom{n-1}{k}+\binom{n-1}{k-1}$
69943
why is ${n+1\choose k} = {n\choose k} + {n\choose k-1}$?
19571
Combinatorial proof for $\sum_{k = 0}^n \binom {r+k} k = \binom {r + n + 1} n$
89552
proving combinatorics identity - $\sum_{k=0}^m{n-k \choose m-k}={n+1 \choose m}$
1125
Inductive proof that ${2n\choose n}=\sum{n\choose i}^2.$
136459
Prove ${n - 1 \choose k - 1} + {n - 2 \choose k - 1} + {n - 3 \choose k - 1} + \dots + {k - 1 \choose k - 1} = {n \choose k}$
1158
How to prove this binomial identity $\sum_{r=0}^n {r {n \choose r}} = n2^{n-1}$?
116461
Prove the identity $\sum_{k=0}^{n}(-1)^{k}\binom{n}{k}=0$ using a combinatorial interpretation of the positive and negative terms
74730
Combinatorial Proof of ${{n}\choose{k}} {{k}\choose{j}} = {{n}\choose{j}} {{n-j}\choose{k-j}}$
6366
Combinatorial proof of summation of $\sum\limits_{k = 0}^n {n \choose k}^2= {2n \choose n}$
133181
Hint on how to prove the identity $\sum_{0 \leq k \leq n } \binom{k}{m} = \binom{n + 1}{m + 1}$
54442
Sum of $k {n \choose k}$ is $n2^{n-1}$
925
Prove that $F_n={n-1 \choose 0 }+{n-2 \choose 1 }+{n-3 \choose 2 }+\ldots$ where $F(n)$ is the $n$-th fibonacci number
13236
Proving equality - a sum including binomial coefficient $\sum_{k=1}^{n}k{n \choose k}2^{n-k}=n3^{n-1}$
31766
Prove that $a_{n}=0$ for all $n$, if $\sum a_{kn}=0$ for all $k\geq 1$
103732
Proving that for $n \equiv 0 \pmod{2}$, we get ${n \choose 0} < {n \choose 1} <\ldots< {n \choose n/2-1}<{n\choose n/2}$ etc.