Count Distinct Preferred-Hat Assignments

### Problem There are n people and hat labels 1..40. Person i must receive one label in hats[i], and no label may be assigned to two people. Count the assignments to the labeled people, modulo 1000000007. ```cpp class Solution { public: int numberWays(vector<vector<int>>& hats); }; ``` ### Examples **Example 1:** ```text Input: hats = [[1,2],[2,3]] Output: 3 Explanation: The ordered assignments are(1,2),(1,3),and(2,3). ``` **Example 2:** ```text Input: hats = [[7],[7]] Output: 0 Explanation: Both people require the same single hat, which cannot be shared. ``` ### Constraints - n equals hats.length, with `1 <= n <= 10`. - Each person lists 1 to 40 unique labels, all in `[1,40]`. ### Contract considerations Only the count modulo the stated modulus is returned, not a selected assignment. ```hint Read the boundary condition Respect each person's preferences and prohibit sharing a hat label. Distinct people remain distinct when their preference lists agree. ``` ### Boundary reminders Only the count modulo the stated modulus is returned, not a selected assignment.