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

Let’s parse the problem: we need plates consisting of two letters followed by three digits, with no repetitions within letters or within digits.
First, choose two distinct letters for the first two positions. The number of ways to arrange two distinct letters from 26 is the permutation 26P2 = 26 × 25 = 650.
Next, choose three distinct digits for the last three positions. The number of ways to arrange three distinct digits from 10 is 10P3 = 10 × 9 × 8 = 720.
Since the letter part and the digit part are independent, multiply the counts: 650 × 720 = 468,000.
Therefore, the total number of license plates is 468,000, which matches the given option a. Note that common mistakes include treating letters or digits as if repetition were allowed, or multiplying by extra factors for order where none are missing, but the no-repetition constraint and fixed positions prevent those errors.
(The provided data lists only one option: a. 468 000; this is consistent with the correct calculation above.)

MUF0092 Mathematics Unit 2 - Semester 2, 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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