A man has 7 relatives. 4 of them are girls and 3 boys. His wife also has 7 relatives, 3 of them are girls and 4 boys. In how many ways can they host a dinner party of 3 girls and 3 boys 50
that there are 3 of man's relatives and 3 of woman's relatives?
Answers
Given:
7 relatives of a man comprises 4 ladies and 3 gentleman his wife his also 7 relatives, 3 of them are ladies and 4 gentlemen.
To find:
In how many ways can they invite a dinner party of 3 ladies and 3 gentlemen so that there are 3 of man's relative and 3 of the wife's relatives?
Solution:
From given, we have,
7 relatives of a man comprises 4 ladies and 3 gentleman his wife his also 7 relatives, 3 of them are ladies and 4 gentlemen.
The possible cases are:
Case 1: A man invites 3 ladies and woman invites 3 gentlemen
3C3 × 3C3 = 1
Case 2: A man invites 3 ladies and woman invites 3 gentlemen
4C3 × 4C3 = 16
Case 3: A man invites 2 ladies, 1 gentleman and woman invites 2 gentlemen, 1 lady
(4C2 × 3C1) × (3C1 × 4C2) = 324
Case 4: A man invites 2 gentlemen, 1 lady and woman invites 2 ladies, 1 gentleman
(4C1 × 3C2) × (3C2 × 4C1) = 144
Therefore, the total number of ways is = 1 + 16 + 324 + 144 = 485
The number of ways people can be invited to a dinner party is 485.