Combinatorial Proof of $\binom{n+1}{k} = (n+1)\binom{n}{k-1}/k$
9522
Proving $k \binom{n}{k} = n \binom{n-1}{k-1}$
109576
Prove that $\binom{m+n}{m}=\sum\limits_{i=0}^m \binom{m}{i}\binom{n}{i}$
58247
Prove $\sum_{k=0}^n \binom{n+k}{k}=\binom{2n+1}{n}$
44484
Combinatorial identity $i{n \choose i } =n {n-1 \choose i - 1}$
152911
Stirling numbers combinatorial proof
25004
Proving $\binom{m}{n} + \binom{m}{n-1} = \binom{m+1}{n}$ algebraically
121279
Proving an Combination formula $ \binom{n}{k} = \binom{n-1}{k}+\binom{n-1}{k-1}$
24588
Proving the combinatorial identity ${n \choose k} = {n-2\choose k-2} + 2{n-2\choose k-1} + {n-2\choose k}$
147910
Prove that $\sum_{n=0}^N\binom{2N-n}N2^n(n+1)=(1+2N)\binom{2N}N$
49066
Give the combinatorial proof of the identity $\sum\limits_{i=0}^{n} \binom{k-1+i}{k-1} = \binom{n+k}{k}$
959
Prove that ${{n}\choose{k}} = {{n - 2}\choose{k}} + 2{{n - 2}\choose{k - 1}} + {{n - 2}\choose{k - 2}}$.
79124
Proving $\binom {n-1}{r-1}=\sum_{k=0}^r(-1)^k\binom r k \binom{n+r-k-1}{r-k-1}$
49771
Prove that $\sum\limits_{k=0}^r\binom{n+k}k=\binom{n+r+1}r$ using combinatoric arguments.
152084
Binomial-Theorem proof
71564
Combinatorial proof of identity involving central binomial coefficients
142534
Proof of $k {n\choose k} = n {n-1 \choose k-1}$ using direct proof
31196
Prove $\binom{n}{a}\binom{n-a}{b-a} = \binom{n}{b}\binom{b}{a}$