A system enforces a stringent password policy requiring a minimum length of $12$ characters Additionally, each password must contain at least one uppercase letter $($A$-$Z$),$ one digit $(0-9),$ and one special character $($~$!,,$ #$,$ $$,\%^,$ &$,*,).$Calculate the total number of possible passwords if the length is exactly $8$

characters.

Calculate the entropy of passwords adhering to this policy, considering the required character set and length constraints.

$-$ Determine the total number of possible passwords complying with this policy. Compute the variance in password swength when the policy is actered to allow longer passwords $($minimum length of $16$ characters$)$ while restricting the character set to only lowercase letters $($a$-$z$).$

$4-$ Discuss the significance of entropy and variance in relation to the security implications of the system's password policy against brute$-$force attacks.