A number plate is formed with two alphabets followed by three digits, with no…

Concept: This is an application of the Fundamental Counting Principle — when a plate is built from several independent groups of positions and repetition is barred within a group, each successive position in that group has one fewer available choice than the position before it; the total number of ways is the product of the choices at every position across every group.

1. The first alphabet position can be filled in 26 ways (all letters of the alphabet are available).

2. Since repetition is not allowed, the second alphabet position has only 25 letters left to choose from.

3. The first digit position can be filled in 10 ways (digits 0 through 9 are all available).

4. The second digit position then has 9 digits left, since the digit already used cannot repeat.

5. The third digit position has 8 digits left, for the same reason.

6. Multiplying the count at every position together: 26 × 25 × 10 × 9 × 8 = 468000.

Cross-check: Treat the two groups as separate permutations: arranging 2 letters out of 26 without repetition gives P(26, 2) = 26 × 25 = 650, and arranging 3 digits out of 10 without repetition gives P(10, 3) = 10 × 9 × 8 = 720. Multiplying the two independent counts, 650 × 720 = 468000, which matches the direct calculation.