Holiday Contest 2025 - Individual Round - Problem 2

It's Christmas! As he is financially challenged, Bob can only afford to hang 12 Christmas lights, 6 of which are red and 6 of which are green. He randomly attaches his lights to a string and places them between his house and his neighbor's house. Bob has a merry Christmas if in his string of 6 lights (the ones closer to his house), the red lights are all adjacent and the green lights are all adjacent. The probability that Bob has a merry Christmas can be written as \frac{m}{n}, where m and n are relatively prime positive integers. Compute m+n.