$\sum_{k=0}^n \binom{n}{k}x^k(1-x)^{n-k}k$?
4931
variation of the binomial theorem
8806
Summation of binomial coefficients
125712
Working with binomial coefficient $\sum_{k=0}^n (-1)^k \binom nk=0$
44111
Find the value of $\sum_{k=1}^{n}k\binom{n}{k}$?
136580
Proving $\sum_{k=0}^n\binom{2n}{2k} = 2^{2n-1}$
143365
Is $\sum_{i=1}^{n-1}i=\binom{n}{2}$?
107071
Prove $\sum_{k= 0}^{n} k \binom{n}{k} = n \cdot 2^{n - 1}$ using the binomial theorem
64378
Proving a binomial sum identity $\sum _{k=0}^n \binom nk \frac{(-1)^k}{2k+1} = \frac{(2n)!!}{(2n+1)!!}$
42736
Is there a closed form for $\sum_{k=0}^n \frac{x^k}{k!}$?
30672
Show that $\sum\limits_{k=0}^n\binom{2n}{2k}^{\!2}-\sum\limits_{k=0}^{n-1}\binom{2n}{2k+1}^{\!2}=(-1)^n\binom{2n}{n}$
110441
Closed form for a formula with a summation over $i\binom{n-i}{k-1}$, and combinatorial proof?
139842
Seeking a combinatorial proof $\sum\limits _{k=0}^n (n-2k)^2\binom{n}{k}=n\times 2^n$
50226
Prove the identity $\sum_{k=0}^{n}\sum_{r=0}^{k} \binom{k}{r} \binom{n}{k} = 3^n$
5332
Find the value of $\sum_0^n \binom{n}{k} (-1)^k \frac{1}{k+1}$
146200
proof by induction: sum of binomial coefficients $\sum_{k=0}^n (^n_k) = 2^n$
17638
Sum with binomial coefficients: $\sum_{k=0}^{n}{2n\choose 2k}$
158774
Series of binomial coefficients: $\sum\limits_{n=k}^{\infty}{\binom nk}x^n=\frac{x^k}{(1-x)^{k+1}}$