Question

Seven relatives of a man comprises four ladies and three gentlemen; his wife has also seven relatives - three of them are ladies and four gentlemen. In how many ways can they invite $3$ ladies and $3$ gentlemen at a dinner party so that there are three man's relatives and three wife's relatives?

Answer 1

The scheme is as follows:

Husband's relative		Wife's relative		Number of seletions
Male (3)	Female (4)	Male (4)	Female (3)
$3$	$0$	$0$	$3$	${}^3{C}_3{\times }^3{C}_3=1$
$2$	$1$	$1$	$2$	${}^3{C}_2{\times }^4{C}_1{\times }^4{C}_1{\times }^3{C}_2=144$
$1$	$2$	$2$	$1$	${}^3{C}_1{\times }^4{C}_2{\times }^4{C}_2{\times }^3{C}_1=324$
$0$	$3$	$3$	$0$	${}^4{C}_3{\times }^4{C}_3=16$
			Total	$485$