n couples are invited to a party with the condition that every husband should be accompanied by his wife. However, a wife need not be accompanied by her husband. The number of different gatherings possible at the party is

Each wife independently decides whether to attend (2 choices each). For each wife who attends, her husband may independently choose to accompany her (2 choices per husband). First choose which subset of n wives attend: C(n, k) for k=0 to n, which sums to 2^n. Then for each attending wife (k of them), choose whether her husband attends: 2^k ways. Summing over k gives 2^n * C(n, k) * 2^k for each k, which equals C(2n, n) * 2^n = (2n)!/(n! * n!) * 2^n. Option A correctly captures this: choose n people from 2n total, then multiply by 2^n for husband choices.