Why does $\sum_{k=0}^{n} (-1)^k {n \choose k} = 0$ if $n\neq 0$ and 1 otherwise?
7083
Alternating sum of binomial coefficients: given $n \in \mathbb N$, prove $\sum^n_{k=0}(-1)^k {n \choose k} = 0$
92784
Intuitive (combinatorial) proof of $2^n=\sum_{k=0}^n {n\choose k}$
87479
Closed form for $\sum_{k=0}^{n-p}\binom{n}{k}\binom{n}{p+k}$
44111
Find the value of $\sum_{k=1}^{n}k\binom{n}{k}$?
96700
How is $\dfrac1{(1-x)^5}=\sum_{n\geq0}{n+4\choose4}x^n$
68080
Evaluation of $\sum_{k=0}^n{n\choose k}^2u^k$
31766
Prove that $a_{n}=0$ for all $n$, if $\sum a_{kn}=0$ for all $k\geq 1$
140877
Evaluate the sum $\sum_{k=1}^n {n \choose k-1} {n \choose k}$
69498
What is $\sum_{r=0}^n \frac{(-1)^r}{\binom{n}{r}}$?
114575
If $\sum_{i=1}^k \dim F_i > n(k-1)$ then $\cap_{i=1}^k F_i \neq \{0\}$
30672
Show that $\sum\limits_{k=0}^n\binom{2n}{2k}^{\!2}-\sum\limits_{k=0}^{n-1}\binom{2n}{2k+1}^{\!2}=(-1)^n\binom{2n}{n}$
37650
Find $\sum_{m=0}^n\ (-1)^m m^n {n \choose m}$
63879
Prove that $\sum_{k=0}^r {m \choose k} {n \choose r-k} = {m+n \choose r}$
128200
Sequence sum question: $\sum_{n=0}^{\infty}nk^n$
133905
Intuitive proof of $\sum_{k=1}^{n} \binom{n}{k} k^{k-1} (n-k)^{n-k} = n^n$
8360
$\sum_i x_i^n = 0$ for all $n$ implies $x_i = 0$
1167
How to prove that $\sum_{k=0}^{n}(-1)^k \binom {n}{k}=0 $