How many different licence plates can be made using two letters followed by three digits, if neither the letters nor the digits can be repeated?

Question

BlackTom AI · Accepted Answer

We need to count licence plates in the format: two letters followed by three digits, with no repetitions within letters or digits.

Step 1: Count possibilities for the letters. There are 26 letters available, and because letters cannot repeat, the first letter can be any of 26, and the second can be any of the remaining 25. So number of letter combinations = 26 × 25 = 650.

Step 2: Count possibilities for the digits. There are 10 digits (0–9), and digits cannot repeat, so the first digit can be any of 10, the second any of the remaining 9, and the third any of the remaining 8. So number of digit combinations = 10 × 9 × 8 = 720.

Step 3: Combine choices. Since the letter block and the digit block are independent, multiply the counts: 650 × 720.

Step 4: Compute the product. 650 × 720 = (65 × 72) × 100 = 4680 × 100 = 468,000.

Now evaluate the options:
- Option a: 492,804. This would not match the calculated 468,000; it likely comes from a different counting that doesn’t enforce the same non-repetition constraints.
- Option b: 468,000. This exactly matches the product 650 × 720, so it is the correct total under the given conditions.
- Option c: 676,000. Note that 676 is 26 × 26, which ignores the no-repeat constraint for letters; multiplying by digits would give a much larger or mis-scaled total.
- Option d: 1,676. This is far too small for two-letter permutations followed by three-digit permutations with no repeats.
- Option e: 82. This is far off the scale and neglects the combinatorial possibilities for both letters and digits.

Summary: The correct approach yields 650 letter combos times 720 digit combos, giving 468,000 total plates. The plausible, exact match among the choices is option b, 468 000.

How many different licence plates can be made using two letters followed by three digits, if neither the letters nor the digits can be repeated?

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