Suppose that a password must have at least $8$, but not more than $10$ characters, where each character in the password is either one of the $26$ lower case English letters, or one of the $10$ digits, or one of the $6$ special characters $∗, \gt, \lt, !, +, =$.
$(i)$ How many different passwords are there?
$(ii)$ How many of these passwords contain at least one occurrence of at least one of the special characters?
For part $(i)$, I think it's a total of $42$ characters therefore $42\choose10$ (no. of $10$ character passwords) minus $42\choose8$ (no. of $8$ character passwords) . For part $(ii)$, I've no idea.
I'd appreciate a bit of guidance.