How many bijections $f : \{1, 2, 3, 4, 5\} \to \{1, 2, 3, 4, 5\}$ have $f(1) \ne 1$?

Explanation

1. 01

   There are $5! = 120$ bijections; those with $f(1) = 1$ arrange the other 4 elements: $4! = 24$.

2. 02

   $120 - 24 = 96$; or 4 choices for $f(1)$, then $4! = 24$ for the rest: $4 \times 24 = 96$.