A man has 7 relatives,4 of them are ladies an | vedclass

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(C) Let $m_L, m_G$ be the number of ladies and gents invited from the man's relatives,and $w_L, w_G$ be the number of ladies and gents invited from th

Vedclass · Accepted Answer

$485$

Vedclass · Answer

(C) Let $m_L, m_G$ be the number of ladies and gents invited from the man's relatives,and $w_L, w_G$ be the number of ladies and gents invited from the wife's relatives.We need $m_L + w_L = 3$ and $m_G + w_G = 3$,where $0 \le m_L, m_G \le 3$ and $0 \le w_L, w_G \le 3$.The man has $4$ ladies and $3$ gents. The wife has $3$ ladies and $4$ gents.The possible cases for $(m_L, m_G)$ and $(w_L, w_G)$ are:$1$. $(m_L, m_G) = (0, 3)$ and $(w_L, w_G) = (3, 0)$: Ways $= {^4C_0} 	imes {^3C_3} 	imes {^3C_3} 	imes {^4C_0} = 1 	imes 1 	imes 1 	imes 1 = 1$.$2$. $(m_L, m_G) = (1, 2)$ and $(w_L, w_G) = (2, 1)$: Ways $= {^4C_1} 	imes {^3C_2} 	imes {^3C_2} 	imes {^4C_1} = 4 	imes 3 	imes 3 	imes 4 = 144$.$3$. $(m_L, m_G) = (2, 1)$ and $(w_L, w_G) = (1, 2)$: Ways $= {^4C_2} 	imes {^3C_1} 	imes {^3C_1} 	imes {^4C_2} = 6 	imes 3 	imes 3 	imes 6 = 324$.$4$. $(m_L, m_G) = (3, 0)$ and $(w_L, w_G) = (0, 3)$: 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,$4$ of them are ladies and $3$ are gents; his wife has $7$ other relatives,$3$ of them are ladies and $4$ are gents. The number of ways they can invite $3$ ladies and $3$ gents to a party such that there are $3$ of the man's relatives and $3$ of the wife's relatives,is:

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