A regular cube has its 12 edges colored with 12 different colors. Each edge is colored with exactly one color, and any two edges that share a vertex must receive different colors. Express the probability that all 12 edges are colored with pairwise distinct colors as a reduced fraction and calculate .
This problem is based on RUC-AIBOX. OlymMATH: A Dataset of Olympiad-Level Math Problems (en-hard split, test row 31). Available at: https://huggingface.co/datasets/RUC-AIBOX/OlymMATH
[THOUGHT] The problem involves coloring the 12 edges of a cube with 12 distinct colors such that adjacent edges (sharing a vertex) have different colors. The key insight is that if all 12 edges are colored with pairwise distinct colors, the adjacency condition is automatically satisfied because no two edges share the same color. Therefore, any permutation of the 12 colors on the edges is a valid coloring. The total number of such permutations is 12!, and all of them are valid. Hence, the probability is 1, leading to the fraction 1/1. The sum of the numerator and denominator is 1 + 1 = 2.
[PYTHON]
The probability is 1, so the fraction is 1/1. Thus, p + q = 1 + 1 = 2.
print(2) [/PYTHON]
[THOUGHT] The problem requires finding the probability that all 12 edges of a cube are colored with pairwise distinct colors, given that adjacent edges (sharing a vertex) must have different colors. Since the cube has exactly 12 edges and there are 12 distinct colors, any assignment of a unique color to each edge automatically satisfies the adjacency condition. Therefore, all 12! permutations of the colors are valid. The probability is thus 1, which simplifies to the fraction 1/1. Consequently, p + q = 1 + 1 = 2. [/THOUGHT] \boxed{2}