GATE CS 2003 Q5 — Combinatorics: Counting, Recurrence Relations, Generating Functions

To find the number of different gatherings possible, we can analyze the choices available for each of the $n$ couples independently.

For any single couple consisting of a husband ($H$) and a wife ($W$), let's look at the possible attendance scenarios:

1. Neither attends: This is valid because the husband is not present, so the condition of him being accompanied by his wife is not violated.
2. Only the wife ($W$) attends: This is valid because a wife does not need to be accompanied by her husband.
3. Both ($H$ and $W$) attend: This is valid because the husband is accompanied by his wife.
4. Only the husband ($H$) attends: This is invalid because the husband must be accompanied by his wife.

Thus, there are exactly $3$ valid choices for each couple.

Since the decision for each of the $n$ couples is independent of the others, we apply the product rule of combinatorics:

Hence, the correct option is B.