Question

There are four balls of different colours and four boxes of same colours in which the balls, one each in a box could be placed such that a ball does not go to a box of its own colour. The number of ways this can be done is

• A. 9
• B. 18
• C. 12
• D. None of these

Answer 1

Answer: (A)

9

Answer 2

This question is based on de arrangement theory which states that “If n things are arranged in a row, the number of ways in which they can be deranged so that no one of them occupies is original place is

$n!\mspace{6mu}\left\{ 1 - \frac{1}{11} + \frac{1}{2!} - \frac{1}{3!} + ... + ( - 1)\frac{n!}{n!} \right\}$

Here n = 4,

∴ Required ways = 4! $\left( 1 - \frac{1}{1!} + \frac{1}{2!} - \frac{1}{3!} + \frac{1}{4!} \right)$

= 24 $\left( \frac{1}{2} - \frac{1}{6} + \frac{1}{24} \right)$

= 12 – 4 + 1 = 9