dp[i][mask] = Number of ways to match first i men with the set of women indicated by the mask, 0 <= i <= n, 0 <= mask <= 2^n. Obviuosly, if i != popcount(mask), dp[i][mask] = 0.

Fill the dp table by iterating over i and mask. When you fill dp[i][mask], you iterate over all possible pairing of i-th man with a woman. So the transition is dp[i][mask] = (sum of dp[i-1][mask ^ 1 << j] where j-th woman appeared in the mask). Code

You can refer to explanation by FlowerOfSorrow(above comment) and this code — https://atcoder.jp/contests/dp/submissions/15835680