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

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Question

(B) Let $P$ invite $l_1$ ladies and $m_1$ men from his friends,and $Q$ invite $l_2$ ladies and $m_2$ men from her friends.Given that $P$ invites $3$ f

Vedclass · Accepted Answer

$485$

Vedclass · Answer

(B) Let $P$ invite $l_1$ ladies and $m_1$ men from his friends,and $Q$ invite $l_2$ ladies and $m_2$ men from her friends.Given that $P$ invites $3$ friends,so $l_1 + m_1 = 3$.Given that $Q$ invites $3$ friends,so $l_2 + m_2 = 3$.The total number of ladies invited is $l_1 + l_2 = 3$.The total number of men invited is $m_1 + m_2 = 3$.We have the following cases:Case $1$: $l_1=3, m_1=0$ and $l_2=0, m_2=3$. Number of ways = $^4C_3 	imes ^3C_0 	imes ^3C_0 	imes ^4C_3 = 4 	imes 1 	imes 1 	imes 4 = 16$.Case $2$: $l_1=2, m_1=1$ and $l_2=1, m_2=2$. Number of ways = $^4C_2 	imes ^3C_1 	imes ^3C_1 	imes ^4C_2 = 6 	imes 3 	imes 3 	imes 6 = 324$.Case $3$: $l_1=1, m_1=2$ and $l_2=2, m_2=1$. Number of ways = $^4C_1 	imes ^3C_2 	imes ^3C_2 	imes ^4C_1 = 4 	imes 3 	imes 3 	imes 4 = 144$.Case $4$: $l_1=0, m_1=3$ and $l_2=3, m_2=0$. Number of ways = $^4C_0 	imes ^3C_3 	imes ^3C_3 	imes ^4C_0 = 1 	imes 1 	imes 1 	imes 1 = 1$.Total number of ways = $16 + 324 + 144 + 1 = 485$.

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