Summation of Products of n Numbers taken m at a time with Repetitions/Corollary/Examples/12

Examples of Corllary to Summation of Products of n Numbers taken m at a time with Repetitions
Consider the result Summation of Products of n Numbers taken m at a time with Repetitions:


 * $\ds \sum_{a \mathop \le j_1 \mathop \le \cdots \mathop \le j_m \mathop \le b} \left({\prod_{k \mathop = 1}^m x_{j_k} }\right) = \sum_{\substack {k_1, k_2, \ldots, k_m \mathop \ge 0 \\ k_1 \mathop + 2 k_2 \mathop + \cdots \mathop + m k_m \mathop = m} } \left({\prod_{j \mathop = 1}^m \dfrac { {S_j}^{k_j} } {j^{k_j} k_j !} }\right)$

where:
 * $S_r = \displaystyle \sum_{k \mathop = a}^b {x_k}^r$

for $r \in \Z_{> 0}$

Let $m = 12$.

Then one of the summands on the is:
 * $\dfrac {S_1 {S_2}^3 S_5} {240}$

Proof
$12$ can thus be expressed as
 * $12 = 5 + 2 + 2 + 2 + 1$

We have that $k_j$ is the number of instances of $j$ in the partition.

Thus setting $k_1 = 1, k_2 = 3, k_5 = 5$:

Hence the summand: