7 relatives of a man comprises 4 ladies and 3 gentlemen, his wife also has 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

485

Explanation

The possible cases are
Case I A man invites 3 ladies and women invites 3 gentlemen.
Number of ways `= ""^(4)C_(3) * ""^(4)C_(3) = 16`
Case II A man invites (2 ladies, 1 gentleman) and women invites (2 gentlemen, 1 lady).
Number of ways `= (""^(4)C_(2) * ""^(3)C_(1))*(""^(3)C_(1)*""^(4)C_(2)) = 324`
Case III A man invites (1 lady, 2 gentlemen) and women invites (2 ladies, 1 gentleman).
Number of ways `= (""^(4)C_(1) * ""^(3)C_(2))*(""^(3)C_(2)*""^(4)C_(1)) = 144`
Case IV A man invites (3 gentlemen) and women invites (3 ladies).
Number of ways `= ""^(3)C_(3) * ""^(3)C_(3) = 1`
`therefore` Total number of ways,
= 16 + 324 + 144 + 1 = 485