MathNet 2019-part-02 — 184 problems (EN–VI) Question 132

We define the weight of a pair of numbers $\{a, b\}$ as $|a-b|$. In how many ways can the set $\{1, 2, \dots, 12\}$ be divided into six pairs so that the total sum of weights of all pairs equals $30$?

(Japan 2018)