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

We need to select which wives attend (choose n out of 2n women), then each attending wife brings her husband. If k wives are selected, we must include their k husbands, totaling 2k people from k couples. The number of ways to choose k couples from n is nCk, and from these k couples, all 2k people attend. Summing over k from 0 to n: Σ(k=0 to n) nCk = 2^n. Then 2nCn accounts for selecting which n positions/couples.