Working with binomial coefficient $\sum_{k=0}^n (-1)^k \binom nk=0$
7083
Alternating sum of binomial coefficients: given $n \in \mathbb N$, prove $\sum^n_{k=0}(-1)^k {n \choose k} = 0$
142303
Alternating sum of binomial coefficients multiplied by (1/k+1)
116722
Simplifying $\big(\sum_{i=0}^n\binom{k}{i}\binom{M-k}{n-i}\frac{k-i}{M-n}\big)/{\binom{M}{n}}$
122006
Proving combinatorial identity $\sum_{k=0}^{n}(-1)^k \binom{2n-k}k 2^{2n-2k} = 2n+1$ "directly"
83400
Sum of products of binomial coefficients: $ \sum_{l=0}^{n-j} \binom{M-1+l}{l} \binom{n-M-l}{n-j-l} = \binom{n}{j} $
158774
Series of binomial coefficients: $\sum\limits_{n=k}^{\infty}{\binom nk}x^n=\frac{x^k}{(1-x)^{k+1}}$
155769
Prove $\sum_{k= 0}^{n} k \binom{n}{k} = n \cdot 2^{n-1}$ using the binomial theorem for $n\geq1$
44111
Find the value of $\sum_{k=1}^{n}k\binom{n}{k}$?
17638
Sum with binomial coefficients: $\sum_{k=0}^{n}{2n\choose 2k}$
51008
Evaluating a sum involving binomial coefficient in denominator
72639
Show that $\sum_{k=0}^n\binom{3n}{3k}=\frac{8^n+2(-1)^n}{3}$
21000
Calculating $\sum_{k=1}^nk(k!)$ combinatorially
110441
Closed form for a formula with a summation over $i\binom{n-i}{k-1}$, and combinatorial proof?
19914
Closed form for $\sum_{n=0}^{\infty} \binom{n+k}{k} x^n$ (Negative Binomial Theorem)
13236
Proving equality - a sum including binomial coefficient $\sum_{k=1}^{n}k{n \choose k}2^{n-k}=n3^{n-1}$
72026
Induction proof of $\sum_{k=1}^{n} \binom n k = 2^n -1 $
56828
A combinatorial proof that the alternating sum of binomial coefficients is zero