Question

7 relatives of a man comprises 4 ladies and 3 gentlemen; his wife has also 7 relatives; 3 of them are ladies and 4 gentlemen. 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?

Answer 1

The possible cases are 

Case I : A man invites 3 ladies and women invites 3 gentlemen. 

Number of ways = ⁴C₃.⁴C₃ = 16 

Case II : A man invites (2 ladies, 1 gentleman) and women invites (2 gentlemen, 1 lady). 

Number of ways = (⁴C₂.³C₁ ).(³C₁ . ⁴C₂) = 324 

Case III : A man invites (1 lady, 2 gentlemen) and women. invites (2 ladies, 1 gentleman). 

Number of ways = (⁴C₁.³C₂). (³C₂.⁴C₁) = 144 

Case IV : A man invites (3 gentlemen) and women invites (3 ladies). 

N umber of ways = ³C₃ .³C₃ = 1

.'. Total number of ways =16 + 324 + 144 + 1 = 485