A man has 7 relatives (4 women and 3 men). His wife also has 7 relatives (3 women and 4 men). In how many ways can they invite 3 women and 3 men so that 3 of them are man's relatives and 3 of them are his wife's relatives?

Let i = number of man's women relatives invited (0 to 3), then man's men relatives = 3−i. Wife's women = 3−i, wife's men = 3−(3−i) = i. Number of ways = Σ C(4,i)·C(3,3−i)·C(3,3−i)·C(4,i) for i=0,1,2,3. i=0: 1·1·1·1 = 1. i=1: 4·3·3·4 = 144. i=2: 6·3·3·6 = 324. i=3: 4·1·1·4 = 16. Total = 1+144+324+16 = 485.