A man has 7 relatives: 4 ladies and 3 gentlem | vedclass

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Question

(A) Let the number of ladies and gentlemen invited from the man's relatives be $(L_m, G_m)$ and from the wife's relatives be $(L_w, G_w)$.We need $L_m

Vedclass · Accepted Answer

$485$

Vedclass · Answer

(A) Let the number of ladies and gentlemen invited from the man's relatives be $(L_m, G_m)$ and from the wife's relatives be $(L_w, G_w)$.We need $L_m + L_w = 3$ and $G_m + G_w = 3$,where $L_m + G_m = 3$ and $L_w + G_w = 3$.The possible cases for $(L_m, G_m)$ and $(L_w, G_w)$ are:Case $1$: $(L_m, G_m) = (0, 3)$ and $(L_w, G_w) = (3, 0)$.Ways $= {^4C_0} 	imes {^3C_3} 	imes {^3C_3} 	imes {^4C_0} = 1 	imes 1 	imes 1 	imes 1 = 1$.Case $2$: $(L_m, G_m) = (1, 2)$ and $(L_w, G_w) = (2, 1)$.Ways $= {^4C_1} 	imes {^3C_2} 	imes {^3C_2} 	imes {^4C_1} = 4 	imes 3 	imes 3 	imes 4 = 144$.Case $3$: $(L_m, G_m) = (2, 1)$ and $(L_w, G_w) = (1, 2)$.Ways $= {^4C_2} 	imes {^3C_1} 	imes {^3C_1} 	imes {^4C_2} = 6 	imes 3 	imes 3 	imes 6 = 324$.Case $4$: $(L_m, G_m) = (3, 0)$ and $(L_w, G_w) = (0, 3)$.Ways $= {^4C_3} 	imes {^3C_0} 	imes {^3C_0} 	imes {^4C_3} = 4 	imes 1 	imes 1 	imes 4 = 16$.Total ways $= 1 + 144 + 324 + 16 = 485$.

$A$ man has $7$ relatives: $4$ ladies and $3$ gentlemen. His wife also has $7$ relatives: $3$ ladies and $4$ gentlemen. In how many ways can they invite a dinner party of $3$ ladies and $3$ gentlemen,such that there are $3$ of the man's relatives and $3$ of the wife's relatives?

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