Q. 7 relatives of a man comprise 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 relative?
Explanation
(a)
The possible cases are
Case I A man invites 3 ladies and woman invites 3 gentleman
$\Rightarrow \ \ \ ^{4}C_{3}\ ^{4}C_{3} = 16$
Case II A man invites (2 ladies, 1 gentlemen) and woman invites (2 gentlemen, 1 lady)
$\Rightarrow \ \ \ \left( \ ^{4}C_{2}.\ ^{3}C_{1} \right).\left( \ ^{3}C_{1}.\ ^{4}C_{2} \right) = 324$
Case III A man invites (1 lady, 2 gentlemen) and woman invites (2 ladies, 1 gentlemen)
$\Rightarrow \ \ \ \left( \ ^{4}C_{1}.\ ^{3}C_{2} \right).\left( \ ^{3}C_{2}.\ ^{4}C_{1} \right) = 144$
Case IV A man invites (3 gentlemen) and woman invites (3 ladies)
$\Rightarrow \ \ \ ^{3}C_{3}.\ ^{3}C_{3} = 1$
$\therefore$ Total number of ways
$= 16 + 324 + 144 + 1 = 485$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.