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  4. 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

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Q. 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

Permutations and Combinations

Solution:

(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 $