A maxX 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 have no common friends. Then the total number of ways in which X and Y together can throw a party inviting 3 ladies and 3 men, so that 3 friends of each X and Y are in this party is.

X has 4 ladies friends and 3 men friends. Y has 3 ladies friends and 4 men friends. X and Y have to invite total 3 ladies and 3 men to the party and each of X and Y should bring in 3 people to the party. Consider the cases as (LX, MX, LY, MY) where L and M denote the lady or man corresponding to X and Y. The possible cases are as follows:
Cases No. of ways Numerical Value
(3,0,0,3) 4C3 * 4C3 = 16
(2,1,1,2) 4C2 * 3C1 * 3C1 * 4C2 = 324
(1,2,2,1) 4C1 * 3C2 * 3C2 * 4C1 = 144
(0,3,3,0) 3C3 * 3C3 = 1
Hence, the total number of ways = 16 + 324 + 144 + 1 = 485.