If $P = \{1,2,3,4,5\}$ and $Q = \{0,1,2,3,4,5\}$. The number of one one function function from
$P$ to $Q$ such that $g(1)\neq 0$ and $g(i)\neq i$ for $i=1,2,3,4,5$ is?
My attempt: Since $g(1) \neq 0,1$, $g(1)$ can take any $4$ values as $2,3,4,5$, which can be done in $4$ ways. Now $g(2),g(3),g(4),g(5)$ can take the remaining $5$ values which can be done in $\displaystyle \binom{5}{4}\cdot 4!$ ways. As such, the number of one-one functions such that $g(1)\neq0,1$ equals:
$$\displaystyle 4\cdot \binom{5}{4}\cdot 4! = 480$$
How can I find the number of functions for which $g(i) \neq i$ for $i = 2, 3, 4, 5$?