2022 AIME I Problem 9

Ellina has twelve blocks, two each of red (\textbf{R}), blue (\textbf{B}), yellow (\textbf{Y}), green (\textbf{G}), orange (\textbf{O}), and purple (\textbf{P}). Call an arrangement of blocks \textit{even} if there is an even number of blocks between each pair of blocks of the same color. For example, the arrangement \textbf{R B B Y G G Y R O P P O} is even. Ellina arranges her blocks in a row in random order. The probability that her arrangement is even is \frac{m}{n}, where m and n are relatively prime positive integers. Find m+n.

Leading zeroes must be inputted, so if your answer is 34, then input 034

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.