Question
Aman $$X$$ 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 $$Y$$ 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 of $$X$$ and $$Y$$ are in this party, is:
A.
484
B.
485
C.
468
D.
469
Answer :
485
Solution :
Possible cases for $$X$$ are
(1) 3 ladies, 0 man
(2) 2 ladies, 1 man
(3) 1 lady, 2 men
(4) 0 ladies, 3 men
Possible cases for $$Y$$ are
(1) 0 ladies, 3 men
(2) 1 lady, 2 men
(3) 2 ladies, 1 man
(4) 3 ladies, 0 man
No. of ways $$^4{C_3}.{\mkern 1mu} {{\mkern 1mu} ^4}{C_3} + {\left( {^4{C_2}.{\mkern 1mu} {{\mkern 1mu} ^3}{C_1}} \right)^2} + {\left( {^4{C_1}.{\mkern 1mu} {{\mkern 1mu} ^3}{C_2}} \right)^2} + {\left( {^3{C_3}} \right)^2}$$
$$= 16 + 324 + 144 + 1 = 485$$
Possible cases for $$X$$ are
(1) 3 ladies, 0 man
(2) 2 ladies, 1 man
(3) 1 lady, 2 men
(4) 0 ladies, 3 men
Possible cases for $$Y$$ are
(1) 0 ladies, 3 men
(2) 1 lady, 2 men
(3) 2 ladies, 1 man
(4) 3 ladies, 0 man
No. of ways $$^4{C_3}.{\mkern 1mu} {{\mkern 1mu} ^4}{C_3} + {\left( {^4{C_2}.{\mkern 1mu} {{\mkern 1mu} ^3}{C_1}} \right)^2} + {\left( {^4{C_1}.{\mkern 1mu} {{\mkern 1mu} ^3}{C_2}} \right)^2} + {\left( {^3{C_3}} \right)^2}$$
$$= 16 + 324 + 144 + 1 = 485$$