Question

$7$ relatives of a man comprises $4$ ladies and $3$ gentleman his wife his 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 relatives?

• A. $485$
• B. $500$
• C. $486$
• D. $102$

Answer 1

The correct option is A $485$

The possible cases are:
Case I : A man invites (3 ladies) and woman invites (3 gentlemen)
${{}^{4}C}_3{{}^{4}C}_3=16$
Case II : A man invites (2 ladies,1 gentleman) and woman invites (2 gentleman, 1 lady)
$\left({{}^{4}C}_2{{}^{3}C}_1\right)\left({{}^{3}C}_1{{}^{4}C}_2\right)=324$
Case III : A man invites (1 lady, 2 gentlemen) and woman invites (2 ladies, 1 gentleman)
$\Rightarrow \left({{}^{4}C}_1{{}^{3}C}_2\right)\left({{}^{3}C}_2{{}^{4}C}_1\right)=144$
Case IV : A man invites (3 gentlemen) and woman invites (3 ladies)
${{}^{3}C}_3{{}^{3}C}_3=1$
Total number of ways
$=16+324+144+1=485$