There are 5 different coloured balls and 5 boxes of the same colour as that of the balls. The number of ways that exactly 2 balls will go to the boxes of their own colour is

Given 2 balls go into their own colour boxes

$\therefore$  Remaining 3 balls go to wrong colour boxes

$\therefore$  Required number of ways ${=}^n{P}_r\left[1-\frac{1}{1!}+\frac{1}{2!}-\cdots +{\left(-1\right)}^r\frac{1}{r!}\right]$

${=}^5{P}_3\left[1-\frac{1}{1!}+\frac{1}{2!}-\frac{1}{3!}\right]$

$=60.\left(\frac{2}{6}\right)$

$=20$