To determine how many valid passwords can be created with a length of 17, where each character is either a digit or a lowercase letter, we can approach the problem by calculating the number of possible choices for each character in the password.

Step 1: Determine the possible characters

• Digits: There are 10 digits (0 through 9).
• Lowercase letters: There are 26 lowercase letters (a through z).

Thus, for each position in the password, there are $10 + 26 = 36$ possible characters to choose from.

Step 2: Calculate the total number of passwords

Since the password length is 17, and each of the 17 positions can have 36 possible characters, the total number of valid passwords is given by:

$36^{17}$

Let’s compute that.

$36^{17} \approx 1.477201e+26$

So, the total number of valid passwords is approximately $1.477 \times 10^{26}$.

Final Answer:

There are approximately $1.477 \times 10^{26}$ valid passwords of length 17, where each character is either a digit or a lowercase letter.

Would you like further details on how this was calculated or any clarifications?

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Here are 5 related questions that build on this:

1. How would the total number of passwords change if the password length was increased to 20?
2. What if uppercase letters were allowed? How many valid passwords would there be for length 17?
3. How does the number of passwords change if at least one digit is required?
4. What is the probability of randomly generating a password with no repeating characters?
5. If we only allow alphanumeric characters, excluding certain letters like 'o' and 'l' to avoid confusion with numbers, how would the total number change?

Tip: For very large calculations involving powers, logarithms can often help simplify the process.