Problem #318
Not My Scarf Not My Cap
There are 2023 men each having a cap and a scarf of their own. A woman collects caps and scarves of all these n men and distributes each of them a cap and scarf randomly. Find the probability that for every natural number k<1011 there does not exist a subgroup of men of size k such that every man's scarf and cap belongs to someone in the subgroup.
The answer will be p/q (where p and q are co-prime), report it as p \cdot (q^{-1}) \mod (10^9 + 7).