Question

There are 6 different balls and 6 different boxes of the colour same as of the colour of balls then match the column

List-I		List-II
P	The number of ways in which no ball goes in the box of its own colour is	1	16
Q	The number of ways in which atleast 4 balls goes into their own boxes is	2	40
R	The number of ways in which atmost 2 balls goes into their own boxes is	3	265
S	The number of ways in which exactly 3 balls goes into their own boxes is	4	664

• A. P-4, Q-1, R-2, S-3
• B. P-3, Q-1, R-2, S-4
• C. P-4, Q-1, R-3, S-2
• D. P-3, Q-1, R-4, S-2

Answer 1

Answer: (D)

(P) $6!\left[1-\frac{1}{1!}+\frac{1}{2!}-\frac{1}{3!}+\frac{1}{4!}-\frac{1}{5!}+\frac{1}{6!}\right]=265$
[using dearrangement]
(Q) Either four balls go into their correct boxes or all the balls go into the correct boxes
${}^6{C}_4\times 1+1=16$
(R) Either all balls go into incorrect boxes or exactly one ball go into its correct box or two balls go into their correct boxes
$265+{}^6{C}_1\times 5!\left[1-\frac{1}{1!}+\frac{1}{2!}-\frac{1}{3!}+\frac{1}{4!}-\frac{1}{5!}\right]+{}^6{C}_2\times 9$
$265+6\times 44+135=664$
(S) ${}^6{C}_3\times 2$