7 relatives of a man comprise 4 ladies and 3 gentlemen. His wife also has 7 relatives − 3 of them are ladies and 4 are 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 wife’s relative?

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Aicharya · Accepted Answer

$	herefore$ Number of ways to invite a dinner party of 3 ladies and 3 gentlemean, so that there are 3 of man's relative and 3 of wife's relative $ \begin{aligned} & ={ }^3 C_3 \cdot{ }^3 C_3+{ }^3 C_2 \cdot{ }^4 C_1 	imes{ }^4 C_1{ }^3 C_2+{ }^3 C_1{ }^4 C_2 \cdot{ }^4 C_2 \cdot{ }^3 C_1 \ & \ & \quad+{ }^4 C_3 \cdot{ }^4 C_3 \ & =1+3 \cdot 4 \cdot 4 \cdot 3+3 \cdot 6 \cdot 6 \cdot 3+4 \cdot 4 \ & =1+144+324+16 \ & =485 \end{aligned} $

7 relatives of a man comprise 4 ladies and 3 gentlemen. His wife also has 7 relatives − 3 of them are ladies…

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