Prove $\sum_{k=0}^{n}k*p_n(k)=n!$
75227
If $S=\{1,2,\dots,n\}$ and $P_n(k)$ be the number of permutations of $S$ having $k$ fixed points, then $\sum_{k=0}^nk.P_{n}(k)=n!$
1167
How to prove that $\sum_{k=0}^{n}(-1)^k \binom {n}{k}=0 $
133181
Hint on how to prove the identity $\sum_{0 \leq k \leq n } \binom{k}{m} = \binom{n + 1}{m + 1}$
153226
Proof of $\sum_{k = 1}^{n} \frac{1}{k^{2}} < 2 - \frac{1}{n}$
58428
Proving $\sum_{k=0}^{n} {k \choose a} {n-k \choose b} = {n+1\choose a+b+1}$
2455
Prove that $\left(\sum^n_{k=1}x_k\right)\left(\sum^n_{k=1}y_k\right)\geq n^2$
92704
Prove that $\sum_{k=1}^{n}k\binom{n}{k}=n\cdot2^{n-1}$
4991
Proving by induction that $ \sum_{k=0}^n{n \choose k} = 2^n$
7083
Alternating sum of binomial coefficients: given $n \in \mathbb N$, prove $\sum^n_{k=0}(-1)^k {n \choose k} = 0$
4512
Is there a direct, elementary proof of $n = \sum_{k|n} \phi(k)$?
130223
Proving a special case of the binomial theorem: $\sum^{k}_{m=0}\binom{k}{m} = 2^k$
33104
Combinatorial Proof with Summation Identity
14197
show that $\sum_{i=0}^l \dfrac{{l\choose i}}{{n\choose k + i}} = \dfrac{n+1}{(n-l+1) {n-l\choose k}}$
65672
Proof of the summation $n!=\sum_{k=0}^n \binom{n}{k}(n-k+1)^n(-1)^k$?
146200
proof by induction: sum of binomial coefficients $\sum_{k=0}^n (^n_k) = 2^n$
65757
Prove that $n! + k$ is a composite number
107071
Prove $\sum_{k= 0}^{n} k \binom{n}{k} = n \cdot 2^{n - 1}$ using the binomial theorem