MathDB
RBBYGGYROPPO GANGNAM STYLE

Source: AIME I #9

February 9, 2022
AIMEAIME I

Problem Statement

Ellina has twelve blocks, two each of red (R),\left({\bf R}\right), blue (B),\left({\bf B}\right), yellow (Y),\left({\bf Y}\right), green (G),\left({\bf G}\right), orange (O),\left({\bf O}\right), and purple (P).\left({\bf P}\right). Call an arrangement of blocks even if there is an even number of blocks between each pair of blocks of the same color. For example, the arrangement R B B Y G G Y R O P P O {\text {\bf 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 mn\frac mn, where mm and nn are relatively prime positive integers. Find m+nm+n.
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