Prove the identity $\sum_{j=n}^N {j\choose n}={N+1 \choose n+1}$
8476
Proof of the Hockey-Stick Identity: $\sum\limits_{t=0}^n \binom tk = \binom{n+1}{k+1}$
14197
show that $\sum_{i=0}^l \dfrac{{l\choose i}}{{n\choose k + i}} = \dfrac{n+1}{(n-l+1) {n-l\choose k}}$
48446
Summation involving a factorial: $1 + \sum_{j=1}^{n} j!j$
122006
Proving combinatorial identity $\sum_{k=0}^{n}(-1)^k \binom{2n-k}k 2^{2n-2k} = 2n+1$ "directly"
38995
Induction proof that $\sum_{j=n}^{2n-1} (2j + 1) = 3n^2$ - what happened?
106008
Combinatorial Identity $(n-r) \binom{n+r-1}{r} \binom{n}{r} = n \binom{n+r-1}{2r} \binom{2r}{r}$
92784
Intuitive (combinatorial) proof of $2^n=\sum_{k=0}^n {n\choose k}$
111098
Prove $2^{2n} = \sum_{k=0}^{n}\binom{2n+1}{k}$
74887
Combinatorial proof of the identity ${n+1\choose k+1}={k\choose k}+{k+1\choose k}+...+{n-1\choose k}+{n\choose k}$
10917
Give a combinatorial proof: $ n(n+1)2^{n-2} = \sum_{k=1}^{n}k^2\binom{n}{k} $
44328
Prove that $\sum_{k=1} ^ n k^{-1} = \ln(n) + O(1)$
50226
Prove the identity $\sum_{k=0}^{n}\sum_{r=0}^{k} \binom{k}{r} \binom{n}{k} = 3^n$
120976
Proving $\sum_{k=0}^{n}k{n\choose k}^2 = n{2n-1 \choose n-1} $
1139
Combinatorial proof of $\sum_{j=0}^{k} \binom{n}{j} = \sum_{j=0}^k \binom{n-1-j}{k-j}2^j$
37650
Find $\sum_{m=0}^n\ (-1)^m m^n {n \choose m}$
65983
Combinatorial proof of binomial coefficient summation identity
10777
Proof of the identity $2^n = \sum\limits_{k=0}^n 2^{-k} \binom{n+k}{k}$