Using combinatorics to show that $x^{\overline n}=\sum_{k=0}^{n}{ n\brack k}x^k$
112573
Combinatorial proof for Stirling number of 1st kind's generating function
24588
Proving the combinatorial identity ${n \choose k} = {n-2\choose k-2} + 2{n-2\choose k-1} + {n-2\choose k}$
70282
Help finding a combinatorial proof of $k {n \choose k } = n {n - 1 \choose k -1}$
113321
Hint proving this $\sum_{k=0}^{n}\binom{2n}{k}k=n2^{2n-1}$
68080
Evaluation of $\sum_{k=0}^n{n\choose k}^2u^k$
30672
Show that $\sum\limits_{k=0}^n\binom{2n}{2k}^{\!2}-\sum\limits_{k=0}^{n-1}\binom{2n}{2k+1}^{\!2}=(-1)^n\binom{2n}{n}$
143113
Non-inductive, not combinatorial proof of $\sum_{i \mathop = 0}^n \binom n i^2 = \binom {2 n} n$
28027
Find $\sum_{k=0}^{n}4^k \binom{n}{k}$
5049
Proving $\sum_{k=0}^n(-1)^k\binom nk=0$
1837
Showing that $1^k+2^k + \dots + n^k$ is divisible by $n(n+1)\over 2$
109576
Prove that $\binom{m+n}{m}=\sum\limits_{i=0}^m \binom{m}{i}\binom{n}{i}$
59252
Show $\sum_{k=0}^{m} \binom{n-k}{m-k}=\binom{n+1}{m}$
11612
How to show that $\sum_{k=1}^n k(n+1-k)=\binom{n+2}3$?
19424
Showing $(M^\perp)^\perp=\overline{M}$
63879
Prove that $\sum_{k=0}^r {m \choose k} {n \choose r-k} = {m+n \choose r}$
96585
Sums and permutation
75910
Combinatorial Proof of Combinatorial Identity involving $(-1)^k \binom {n-1}{k}$