Question

A man has 14 friends out of which 5 are married couples and 4 are single people. He goes out for dinner with 3 of his friends every night. If he invites one member of a married couple, then he also invites the other. He stops going out for dinner when he has gone out with all possible combinations of his friends. How many times does he go out for dinner?
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Answer 1

The man can only go out to dinner with one couple and one single person or with three single people.

$\therefore$ The number of ways in which he can go out for a dinner with one couple (out of 5) and one single person (out of 4) = ${}^5$C${}_1$ $\times$ ${}^4$C${}_1$ = 20

$\therefore$ The number of ways in which he can go out for a dinner with three single people (out of 4) = ${}^4$C${}_3$ = 4

$\therefore$ The total number of times he goes out for a dinner = 20 + 4 = 24