To determine how many valid passwords there are of length 8, where each character is either a digit $[0-9]$ or a lowercase letter $[a-z]$, with the restriction that the password cannot start with a digit, follow these steps:

1. Number of choices for the first character:
   The first character cannot be a digit, so it must be a lowercase letter. There are 26 lowercase letters in the alphabet $[a-z]$, which means you have 26 choices for the first character.

2. Number of choices for each of the remaining 7 characters:
   For each of the remaining 7 characters, there are no restrictions. Each character can be either a digit or a lowercase letter. There are 10 digits $[0-9]$ and 26 lowercase letters $[a-z]$, which means you have $10 + 26 = 36$ choices for each of the remaining 7 characters.

3. Total number of valid passwords:
   Multiply the number of choices for the first character by the number of choices for each of the remaining characters: $26 \times 36^7$

Now let's calculate the total number of valid passwords.

$26 \times 36^7 = 26 \times 78,364,164,096 = 2,037,468,266,496$

Thus, the total number of valid passwords is $2,037,468,266,496$.

Do you want more details or have any questions?

Here are 5 related questions for further exploration:

1. What if the password length was 6 instead of 8 with the same restrictions?
2. How many valid passwords are there if the first character cannot be a letter instead of a digit?
3. What is the probability of randomly generating a password that starts with the letter 'a'?
4. How many valid passwords are there if no character can be repeated?
5. How does the total number of passwords change if uppercase letters are allowed as well?

Tip: When calculating large powers or combinations, breaking the problem into smaller parts can simplify the calculations.