Question

A man has 7 relatives,4 of them are ladies and 3 of them are gentleman, his wife has also 7 relatives 4 of them are gentleman and 3 of them are ladies.In how many ways can they invite a dinner party for 3 ladies and 3 gentlemen so that there are 3 of the man's relatives and 3 of the wife' relatives.

Answer 1

There four possibilities:
(i) 3 ladies from husband’s side and 3 gentlemen from wife’s side.
No. of ways in this case
= ⁴ C₃ x ⁴ C₃ = 4 x 4 = 16
(ii) 3 gentlemen from husband’s side and 3 ladies from wife’s side.
No. of ways in this case = ³ C₃ x ³ C ₃ = 1 x 1 = 1
(iii) 2 ladies and one gentleman from husband’s side and lady and 2 gentlemen from wife’s side.
No. of ways in this case
= (⁴ C₄ x ³ C₁) x (³ C₁ x ⁴ C₂) = 6 x 3 x 3 x 6 = 324
(iv) One lady and 2 gentlemen from husband’s side and 2 ladies and one gentlemen from wife’s side.
No. of ways in this case
= (⁴ C₁ x ³ C₂) x (³ C₂ x ⁴ C₁) = 4 x 3 x 3 x 4 = 1444
Hence the total no. of ways are
= 16 + 1 + 324 + 144 = 485.