Question

$7$ 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 the man's relatives and $3$ of the wife's relatives?

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

Answer 1

The correct option is A

$485$

Explanation for the correct option:

Step 1: List out all possible cases of invitation:

Both Men and Women have to invite their guests each, and there should be $3$ ladies and $3$ gentlemen at the party.

Then, the man can invite $\left(3L,2L\&1g,1L\&2g,3g\right)$ ,and women can invite $(3g,2g\&1L,1g\&2L,3L)$ correspondingly.

Step 2: Solve each case to get the total number of ways:

Case I: A man sends invites to $3$ of his ladies relatives, and the woman sends invites to $3$ of her gentlemen relatives.

${}^{4}C_3·{}^{4}C_3=16$

Case II: A man sends invites to $2$ of his ladies relatives, $1$ of his gentleman's relative and the woman sends invites to $2$ of her gentlemen relatives and $1$ of her lady relative.

$\left({}^{4}C_2·{}^{3}C_1\right)\left({}^{3}C_1·{}^{4}C_2\right)=324$

Case III: A man sends invites to $1$ of his lady relative, $2$ of his gentlemen relatives and the woman sends invites to$1$of her gentlemen relative and $2$ of her ladies relatives.

$\left({}^{4}C_1·{}^{3}C_2\right)\left({}^{3}C_2·{}^{4}C_1\right)=144$

Case IV: A man sends invites to $3$ of his gentlemen relatives and the woman sends invites to $3$ of her ladies relatives.

${}^{3}C_3·{}^{3}C_3=1$

Step 3: Sum up all the cases to get the required number of ways:

Total ways of invitation$\begin{array}{rcl}& =& 16+324+144+1\\ & =& 485\end{array}$

Hence, option (A) is the correct answer.