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

To solve this problem, I will examine how many choices we have for the letters and for the digits, given that neither letters nor digits can be repeated.
Option a: '468 000' — If there are 26 letters available and no repeats, the first letter has 26 options and the second has 25 (since it cannot repeat the first). For digits, there are 10 options for the first digit, then 9, then 8 for the third due to no repetition. Multiplying: 26 × 25 × 10 × 9 × 8 = 650 × 720 = 468,000. This matches the calculated total, so this option is consistent with the problem’s constraints.
Option b: '82' — This is far too small and does not account for permutations of 2 letters followed by 3 digits with no repeats. It ignores the multiplicative structure of choosing letters and digits in sequence and would only arise if you were counting something like a single feature or a negligible subset, which is not the case here.
Option c: '492 804' — While this number is in the same rough neighborhood as the correct count, it would correspond to a miscalculation such as treating digits or letters with a different repetition rule (for example, allowing one set to repeat or miscounting the available options). The exact product 26×25×10×9×8 does not equal 492,804, so this option is incorrect.
Option d: '1676' — This is a very small number, likely reflecting a miscount from treating the problem as if there were only a few choices or by incorrectly limiting the positions. It fails to capture the 5-position structure (2 letters and 3 digits) with non-repetition.
Option e: '676 000' — Although quite large, this undercounts the letter choices (26×25 = 650) or misweights the digit choices. The digits part 10×9×8 = 720 remains, and multiplying with 650 yields 468,000, not 676,000, so this result is inconsistent with the no-repeat rule applied to both groups.

Summarizing the reasoning, the correct approach multiplies the number of ways to choose two distinct letters (26 × 25) by the number of ways to choose three distinct digits (10 × 9 × 8). The product 26 × 25 × 10 × 9 × 8 equals 468,000, which corresponds to option a. The other options fail because they miscount either the letter choices, the digit choices, or both, by undercounting or by applying an inappropriate repetition rule.

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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