Question 13
Ten adults enter a room, remove their shoes, and toss their shoes into a pile. Later, a child randomly pairs each left shoe with a right shoe without regard to which shoes belong together. The probability that for every positive integer $k \lt 5,$ no collection of $k$ pairs made by the child contains the shoes from exactly $k$ of the adults is $\frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m + n.$