Proving $\sum_{k=0}^n {2n \choose 2k}{2k \choose k}2^{2n-2k} = {4n \choose 2n}$
131326
How to find $\sum_{r\ge 0} \binom{n}{r}\binom{n-r}{r} 2^{n-2r}$?
1125
Inductive proof that ${2n\choose n}=\sum{n\choose i}^2.$
108318
counting proof for $\sum_{k=0}^{n-1} 2^{k} = 2^n - 1$
44161
Using induction to prove $\sum_{k=0}^{N-1} 2^k = 2^N-1$
14197
show that $\sum_{i=0}^l \dfrac{{l\choose i}}{{n\choose k + i}} = \dfrac{n+1}{(n-l+1) {n-l\choose k}}$
31766
Prove that $a_{n}=0$ for all $n$, if $\sum a_{kn}=0$ for all $k\geq 1$
145048
Prove $\sum_{k = 0}^{n}(-1)^{n - k} \binom{n}{k} \cdot k^n = n!$ and $\sum_{k = 0}^{n}(-1)^{n - k} \binom{n}{k} \cdot k^m = 0$
87320
Proving identities like $\sum_{k=1}^nk{n\choose k}^2=n{2n-1\choose n}$ combinatorially
89552
proving combinatorics identity - $\sum_{k=0}^m{n-k \choose m-k}={n+1 \choose m}$
20317
Proving $\sum_{k=1}^{n}{(-1)^{k+1} {{n}\choose{k}}\frac{1}{k}=H_n}$
33082
Proving $\sum_{i=0}^n 2^i=2^{n+1}-1$ by induction.
70128
Prove that $(fg)^{(n)} = \sum_{k=0}^n \binom{n}{k}f^{(k)}(x)g^{(n-k)}(x)$
72026
Induction proof of $\sum_{k=1}^{n} \binom n k = 2^n -1 $
44328
Prove that $\sum_{k=1} ^ n k^{-1} = \ln(n) + O(1)$
1158
How to prove this binomial identity $\sum_{r=0}^n {r {n \choose r}} = n2^{n-1}$?
123955
How to compute $\sum\limits_{k=0}^n (-1)^k{2n-k\choose k}$?
130223
Proving a special case of the binomial theorem: $\sum^{k}_{m=0}\binom{k}{m} = 2^k$