Find the number of ways string of numbers (may contain similar items) could be deranged so that a number is not placed in the same place as it or its similar numbers were placed.
For example, $\{0,0,0,1,1,1\}$ could be arranged in only one way and that is $\{1,1,1,0,0,0\}$.
$\{0,0,0,1,1,1,1\}$ cannot be arranged in any way.
$\{1,2,2,14\}$ can be arranged in $2$ ways i.e $\{2,1,14,2\}$, $\{2,14,1,2\}$.
$\{1,1,2,2,14\}$ can be arranged in $5$ ways i.e $\{2,2,1,1,14\}$, $\{14,2,1,1,2\}$, $\{2,2,14,1,1\}$, $\{2,2,1,14,1\}$, $\{2,14,1,1,2\}$.
Please do try and respond. Thanks in advance.