$n$ 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
Possible outcome for a couple:
Thus $3$ possibilities for each couple, so $3*3*3*\cdots *n$ times = $3^n$
Here it is mentioned wife need not come with husband,
This may give 2 outcomes
wife comes with husband
wife comes with other person other than husband
possible outcomes are
husband with wife
and also none
Please clarify @Arjun Sir
@Jarvis You are inventing cases here , which are not there !
wife comes with other person other than husband = Wife not coming with husband !
We are given only 3 cases here !
You might want to include few more cases , if you follow your logic !
Wife comes with Teddy bear :P and so on...