A man X has 7 friends, 4 of them are ladies and 3 are men. His wife Y also has 7 friends, 3 of them are ladies and 4 are men. Assume X and Y has no common friends. Then the total number of ways in which X and Y together can throw a party inviting 3 ladies annd 3 men, so that 3 friends of each of X and Y are in this party, is

Options

A. 485
B. 468
C. 469
D. 484

Explanation

Given information can be displayed as follows:
`{:(,L,M),(X,4,3),(Y,3,4):}`
A party consisting of 3 ladies and 3 men so that 3 friends of each of X and Y are in it, can be thrown in the following ways:
`{:(,,L,M,,,L,M),(I,X,3,0,II,X,2,1),(,Y,0,3,,Y,1,2),(,,L,M,,,L,M),(III,X,1,2,IV,X,0,3),(,Y,2,1,,Y,3,0):}`
So, required number of ways.
`=(""^(4)C_(3)xx""^(3)C_(0))xx(""^(3)C_(0)xx""^(4)C_(3))+(""^(4)C_(2)xx""^(3)C_(1))xx(""^(3)C_(1)xx""^(4)C_(2))`
`+(""^(4)C_(1)xx^(3)C_(2))xx(""^(3)C_(2)xx""^(4)C_(1))+(""^(4)C_(0)xx""^(3)C_(3))xx(""^(3)C_(3)xx""^(4)C_(0))`