CAT 2023 Slot 1 · Quant · Question 21

Count separately by the number of digits, since the leading-digit rule differs.

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

For two-digit numbers, the tens digit has 9 choices, as it cannot be 0.

The units digit may be any digit other than the one already used, so it has 9 choices.

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

For three-digit numbers, the hundreds digit has 9 choices, again excluding 0.

The tens digit has 9 remaining choices and the units digit has 8.

That gives 9\times 9\times 8=648 three-digit numbers.

The number 1000 itself repeats the digit 0 three times, so it does not qualify.

The total is 9+81+648=738 — the fourth option.

Shortcut. The pattern of choices is 9, then 9\times 9, then 9\times 9\times 8: the leading digit always loses the zero, and each later digit loses one more already-used digit but regains the zero. Totalling, 9+81+648=738.