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 the man's relatives and 3 of the wife's relatives?

Answer 1

As per question, there could be four conditions, when a man and his wife invites their 7 relatives.

Condition 1: A man invites 3 ladies and his wife invites 3 gentlemen.

Number of ways = ⁴C₃.⁴C₃ = 16

Condition 2: A man invites (2 ladies, 1 gentleman) and his wife invites (2 gentlemen, 1 lady).

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

Condition 3: A man invites (1 lady, 2 gentlemen) and his wife invites (2 ladies, 1 gentleman).

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

Condition 4: A man invites (3 gentlemen) and his wife invites (3 ladies).

Number of ways = ³C₃ .³C₃ = 1

Therefore, total number of ways =16 + 324 + 144 + 1 = 485

Answer 2

The possible cases are: 

Case I : A man invites (3 ladies) and woman invites (3 gentlemen) 

⁴C₃ ​⁴C₃ ​ =16 

Case II :  A man invites (2 ladies,1 gentleman) and woman invites (2 gentleman, 1 lady) (⁴C₂ ​³C₁)(³C₁ ​⁴C₂)=324

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

⇒(⁴C₁ ​³C₂)(³C₂ ​⁴C₁)=144

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

³C₃³C₃ ​=1 

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