Proving $r \binom{n}{r}=n\binom{n-1}{r-1}$ combinatorially. (Advice on combinatorial proofs in general?)
3786
Help finding a combinatorial proof of $k {n \choose k } = n {n - 1 \choose k -1}\;$
50226
Prove the identity $\sum_{k=0}^{n}\sum_{r=0}^{k} \binom{k}{r} \binom{n}{k} = 3^n$
71797
Combinatorial argument for $\sum\limits_{k=i}^{n}\binom{n}{k}\binom{k}{i} = \binom{n}{i}2^{n-i}$
147910
Prove that $\sum_{n=0}^N\binom{2N-n}N2^n(n+1)=(1+2N)\binom{2N}N$
109576
Prove that $\binom{m+n}{m}=\sum\limits_{i=0}^m \binom{m}{i}\binom{n}{i}$
130543
If $a_n = \sum\limits_{r=0}^n \binom{n}{r} b_r$, prove $(-1)^n b_n = \sum\limits_{s=0}^n \binom{n}{s} (-1)^s a_s$
42763
How to prove Vandermonde's Identity: $\sum_{k=0}^{n}\binom{R}{k}\binom{M}{n-k}=\binom{R+M}{n}$?
74730
Combinatorial Proof of ${{n}\choose{k}} {{k}\choose{j}} = {{n}\choose{j}} {{n-j}\choose{k-j}}$
133905
Intuitive proof of $\sum_{k=1}^{n} \binom{n}{k} k^{k-1} (n-k)^{n-k} = n^n$
47799
Show that the number or $r$ combinations of $X$ which contain no consecutive integers is given by $\binom{n-r+1}{r}$
47827
Prove that $\sum^{n}_{r=0}(-1)^r\cdot \large\frac{\binom{n}{r}}{\binom{r+3}{r}} = \frac{3!}{2(n+3)}$
121279
Proving an Combination formula $ \binom{n}{k} = \binom{n-1}{k}+\binom{n-1}{k-1}$
129304
Alternative combinatorial proof for $\sum\limits_{r=0}^n\binom{n}{r}\binom{m+r}{n}=\sum\limits_{r=0}^n\binom{n}{r}\binom{m}{r}2^r$
152084
Binomial-Theorem proof
113321
Hint proving this $\sum_{k=0}^{n}\binom{2n}{k}k=n2^{2n-1}$
90904
Combinatorial proof of an identity with squared binomial coefficients
1158
How to prove this binomial identity $\sum_{r=0}^n {r {n \choose r}} = n2^{n-1}$?