Summation of Products of n Numbers taken m at a time with Repetitions/Examples/Order 3

Theorem

 * $\displaystyle \sum_{i \mathop = a}^b \sum_{j \mathop = a}^i \sum_{k \mathop = a}^j x_i x_j x_k = \dfrac { {S_1}^3} 6 + \dfrac {S_1 S_2} 2 + \dfrac {S_3} 3$

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