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

The problem asks for the number of licence plates formed by two letters followed by three digits, with no repetition in either the letters or the digits.
First, count the letters: there are 26 letters in the alphabet, and since repetition is not allowed, the first letter can be any of 26, and the second must be different, giving 26 × 25 = 650 possible letter pairs.
Next, count the digits: there are 10 digits (0–9), and no repetition is allowed among the three digits, so the first digit has 10 choices, the second has 9 choices (different from the first), and the third has 8 choices (different from the first two). This yields 10 × 9 × 8 = 720 possible digit triples.
To obtain the total number of plates, multiply the number of letter options by the number of digit options: 650 × 720 = 468,000.
Option a. 468 000: This matches the computed total, reflecting correct application of the permutation counts for both segments.
Option b. 82: This is far too small and suggests a misunderstanding, perhaps treating both parts as if there were only two choices or ignoring place-value structure.
Option c. 676 000: While closer in magnitude, this overcounts or undercounts certain arrangements, and does not align with the correct 650 × 720 product for the nonrepeating constraint.
Option d. 492 804: This number does not correspond to the correct product of 650 and 720, indicating a miscalculation in the choice of available options for letters or digits or in the multiplication step.
Option e. 1676: This is orders of magnitude smaller and typically indicates treating the problem as if repetition were allowed for a single position or miscounting the placement structure entirely.

MUF0092 Mathematics Unit 2-SEM 1-2025 6.5 Test your knowledge of combinations and permutations

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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