The number of arrangments of all digits of `12345` such that at least `3` digits will not come in its position is

Answer

109

Explanation

Total number of ways such that at least 3 digits will not come in its position.
=Derangement of 3 digits +Derangement of 4 digits +Derangement of 5 digits
`= ""^(5)C_(3)xx3!xx(1-(1)/(1!)+(1)/(2!)-(1)/(3!))+""^(5)C_(4)xx4!xx(1-(1)/(1!)+(1)/(2!)-(1)/(3!)+(1)/(4!))+""^(5)C_(5)xx5!xx(1-(1)/(1!)+(1)/(2!)-(1)/(3!)+(1)/(4!)-(1)/(5!))`
`=10xx2+5xx9+44`
`=20+45+44=109`