Question

The total number number of ways in which all the digits of the number $79853$ can be arranged such that no digit will occur in its original position as given in the number is

Answer 1

Given number is $79853$
Number of digits is $5$
Here, we need to calculate $D_5$
as we know that $D_n=n!(1-\frac{1}{1!}+\frac{1}{2!}-\frac{1}{3!}+.....(-1{)}^n\frac{1}{n!})$
Hence, number of derangements $D_5=5!(1-\frac{1}{1!}+\frac{1}{2!}-\frac{1}{3!}+\frac{1}{4!}-\frac{1}{5!})$
$D_5=120\times (1-1+\frac{1}{2}-\frac{1}{6}+\frac{1}{24}-\frac{1}{120})$
$D_5=60-20+5-1$
$D_5=44$