Question

A man has $7$ relatives, $4$ of them are ladies & $3$ gentlemen; his wife has also $7$ relatives, $3$ of them are ladies & $4$ gentlemen. In how many ways can they invite a dinner party of $3$ ladies & $3$ gentlemen so that there are $3$ ofthe man's relatives & $3$ of the wife's relatives.

• A. $485$
• B. $481$
• C. $324$
• D. $484$

Answer 1

Answer: (A)

$485$

Husband $H,$ Wife $W$
Given:

Relatives of husband $\left(H\right)$

(a) Ladies $\left(L_H\right)=4$
(b) Gentlemen $\left(G_H\right)=3$

Relatives of Wife $\left(W\right)$

(a) Ladies $\left(L_W\right)=3$
(b) Gentlemen $\left(G_W\right)=4$

Case 1: Selecting $\left(3L_H\right)$ and $3\left(G_W\right)$ ways: ${}^4{C}_3{}^4{C}_3=16$

Case 2: Selecting $\left(3G_H\right)$ and $3\left(L_W\right)$ ways: ${}^3{C}_3{}^3{C}_3=1$

Case 3: Selecting $(2L_H$ and $1G_H)$ and $(1L_w$ and $2G_w)$ ways: ${}^4{C}_2{}^3{C}_1{}^3{C}_1{}^4{C}_2=324$

Case 4: Selecting $(1L_H$ and $2G_H)$ and $(2L_W$ and $1G_w)$ ways : ${}^4{C}_1{}^3{C}_2{}^3{C}_2{}^4{C}_1=144$

Add all cases we get: $485$ ways