Sorry if this has been beaten to death.

A password can contain capital and lowercase letters, digits, and underscores.

$63^8$ total passwords, correct?

If we add the stipulation that the first character cannot be a digit-- why wouldn't the number of illegal passwords be $10*63^7$, resulting in $63^8-10*63^7$ legal passwords?

• The question for the answer you gave in your (now deleted) comment just now ($53\cdot 63^7+53\cdot 63^6+\dots+53$) is if the password is allowed to be of any length up to 8 characters in length. There are $63^8$ passwords of exactly length $8$. There are $63^7$ passwords of exactly length $7$, etc... giving $63^8+63^7+63^6+\dots+63$ passwords of any length $1$ through $8$. You can use your method to get $10\cdot 63^7+10\cdot 63^6+10\cdot 63^5+\dots+10$ invalid passwords which you need to subtract if you are answering the question for passwords of any length up to 8 rather than only 8. – JMoravitz Mar 15 '18 at 20:42
• Note also: $63^8-10\cdot 63^7 = 53\cdot 63^7$ – JMoravitz Mar 15 '18 at 20:44

Because there are $10$ options for the first character of the password if it starts with a digit; and for the rest of the characters, there are $63$ options still so $10 \cdot 63^7$ is the number of illegal passwords.
Your logic is correct for 8 digit passwords. For each digit, you could have $26*2+10+1=63$ choices. So, without restrictions,total number of passwords would be $63^d$ where d is the number of characters in the password. If the first character cannot be a digit, then we would only have $53$ choices for the first character, which then yields $53*63^{d-1}$ choices.