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CAT 2024 Slot 3 — QA Question 3

Numbers (P&C)Easy

The number of all positive integers up to 500 with non-repeating digits is

Answer & solution

Answer: 378

Solution

Medium

"Non-repeating digits" means no digit appears twice in the number. Count separately by number of digits: 1-digit, 2-digit, and 3-digit numbers (1 to 499), then check 500 by itself.

1

One-digit numbers (1–9). Every single-digit number trivially has non-repeating digits.

N1=9\begin{aligned} &N_1=9 \end{aligned}
2

Two-digit numbers (10–99). Tens place: any of 11–99 (99 ways). Units place: any digit except the one already used, including 00 (99 ways).

N2=9×9⇒ N2=81\begin{aligned} &N_2=9\times 9\\ &\Rightarrow\ N_2=81 \end{aligned}
3

Three-digit numbers (100–499). Hundreds place must be 1,2,3,41,2,3,4 (44 ways). Tens place: any of the remaining 99 digits. Units place: any of the remaining 88 digits.

N3=4×9×8⇒ N3=288\begin{aligned} &N_3=4\times 9\times 8\\ &\Rightarrow\ N_3=288 \end{aligned}
4

The number 500. Digits are 5,0,05,0,0 — the 00 repeats, so 500500 does not qualify. Add the counts.

N=N1+N2+N3⇒ N=9+81+288[(1)+(2)+(3)]⇒ N=378\begin{aligned} &N=N_1+N_2+N_3\\ &\Rightarrow\ N=9+81+288\quad\text{[(1)+(2)+(3)]}\\ &\Rightarrow\ N=378 \end{aligned}
378378
CAT 2024 Slot 3 QA Q3: The number of all positive integers up to 500 with non-repeating digits is — Solution | TheCATExam