CAT 2024 Slot 3 · Quant · Question 14

Count separately by the number of digits, since the constraints differ.

Single-digit numbers run from 1 to 9, and a single digit cannot repeat, so all 9 qualify.

For two-digit numbers, the first digit has 9 choices (1 to 9, as it cannot be 0).

The second digit may be anything except the first, so it has 9 choices from the ten digits.

That gives 9\times 9=81 two-digit numbers.

For three-digit numbers up to 500, the hundreds digit can only be 1, 2, 3 or 4 — that is 4 choices.

The tens digit is any of the remaining 9 digits, and the units digit any of the remaining 8.

That gives 4\times 9\times 8=288 such numbers.

The number 500 itself has two repeated zeros, so it does not qualify and nothing is added for it.

The total is 9+81+288=378.

Shortcut. Counting the hundreds digit first is what keeps the 500 boundary painless — restricting it to \{1,2,3,4\} covers everything from 100 to 499 in one stroke, and 500 is then dismissed by inspection.