There are five balls of different colours and five boxes of colours same as those of the balls. The number of ways in which the balls, one each in a box, could be placed such that exactly one ball goes to a box of its own colour, is

Options

A. 9
B. 24
C. 45
D. 120

Explanation

Let `b_(1),b_(2),......b_(5)` be five balls and `B_(1),B_(2),......,B_(5)` be five boxes. Then,
Requried number of ways
`sum_(r=1)^(5)(r^(th)" ball is placed in "r^(th)" box")xx("Remaining balls are placed in wrong boxes")`
`=sum_(r=1)^(5)1xx4!{-1(1)/(1!)+(1)/(2!)-(1)/(3!)+(1)/(4!)}`
`=sum_(r=1)^(5)(12-4+1)=45`.