1. ## Binomial coefficients?

Let k and n be positive integers and put S(k) = 1^k + 2^k + 3^k + ... + n^k.

Show that (m + 1)C(1)S(1) + (m + 1)C(2)S(2) + ... + (m + 1)C(m)S(m) = (n + 1)^(m + 1) - (n + 1)

NOTE: (m + 1)C(1) is (m + 1) choose 1
S(k) - k is subscript

2. In this post we we'll take: $0^0=1$ to shorten the thing.

Note that: $
\sum\limits_{j = 1}^n {\left[ {\left( {j + 1} \right)^k - j^k } \right]} = \left( {n + 1} \right)^k - 1
$

By the Binomial Theorem: $
\sum\limits_{j = 1}^n {\left[ {\sum\limits_{s = 0}^k {\binom{k}{s}\cdot j^s } - j^k } \right]} =\sum\limits_{j = 1}^n {\left[ {\sum\limits_{s = 0}^{k-1}{\binom{k}{s}\cdot j^s }} \right]} = \left( {n + 1} \right)^k - 1
$

Now reverse the Order of the sums: $\sum\limits_{s = 0}^{k-1} {\binom{k}{s}\cdot\left[ {\sum\limits_{j = 1}^{n}{ j^s }} \right]} = \left( {n + 1} \right)^k - 1
$

Note that for $s=0$: $\binom{k}{s}\cdot \sum\limits_{j = 1}^{n}{ j^s }=n$ thus $\sum\limits_{s = 1}^{k-1} {\binom{k}{s}\cdot\left[ {\sum\limits_{j = 0}^{n}{ j^s }} \right]} = \left( {n + 1} \right)^k - (n+1)
$