To determine the number of possible passwords, let's break down the problem.

1. Digits: We need three distinct digits, chosen from the set $\{0, 1, 2, \ldots, 9\}$. Since digits cannot be repeated, we can calculate the number of ways to pick and arrange these three digits:

   • For the first digit, there are 10 choices.
   • For the second digit, there are 9 remaining choices (since the first digit can't be reused).
   • For the third digit, there are 8 remaining choices.

   So, the total number of ways to arrange three distinct digits is: $10 \times 9 \times 8 = 720$

2. Letters: We need five distinct letters, chosen from the 26 letters of the alphabet. Since letters cannot be repeated:

   • For the first letter, there are 26 choices.
   • For the second letter, there are 25 remaining choices.
   • For the third letter, there are 24 remaining choices.
   • For the fourth letter, there are 23 remaining choices.
   • For the fifth letter, there are 22 remaining choices.

   Thus, the total number of ways to arrange five distinct letters is: $26 \times 25 \times 24 \times 23 \times 22 = 7,893,600$

3. Total Password Combinations: Since the three digits and five letters are combined to form the password, the total number of possible passwords is the product of the two results above: $720 \times 7,893,600 = 5,684,992,000$

Answer

There are 5,684,992,000 possible passwords that can be created.

Would you like a more detailed breakdown of any part of the solution?

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1. How would the answer change if repeated characters were allowed?
2. What if there were four digits and four letters instead?
3. How does using uppercase and lowercase letters affect the total count?
4. What if digits could be reused but letters couldn't?
5. What if a specific digit or letter was prohibited from being used in the password?

Tip: Remember that when items are distinct and repetition isn’t allowed, the total combinations are found by multiplying the possible choices for each position sequentially.