Lor
2023 AIME I
Problem 1
Five men and nine women stand equally spaced around a circle in random order. The probability that every man stands diametrically opposite a woman is 
where 
and 
are relatively prime positive integers. Find 
men: m1, m2, … , m5
women: w1, w2, …, w9
Fix m1 first
total= 13!
favorable cases:
There are 9 choices to place opposite to m1
m2 has 12 options
There are 8 choices to place opposite to m2
….
Solution