Question

The number of natural numbers less than 500 in which no two digits are repeated is:

• A. 374
• B. 376
• C. 378
• D. 380

Hint:

For counting numbers with no repeated digits, use multiplication for each digit place, considering the constraints for each step.

Answer 1

The Correct Option is C

Step 1: Count the number of 1-digit numbers.
There are 9 one-digit numbers: \( 1, 2, 3, \dots, 9 \). Step 2: Count the number of 2-digit numbers.
For a 2-digit number, the first digit can be any of \( 1, 2, \dots, 9 \) (9 choices), and the second digit can be any of the remaining 9 digits (0-9, except the first digit). Thus, there are: \[ 9 \times 9 = 81 \quad \text{2-digit numbers} \] Step 3: Count the number of 3-digit numbers.
For a 3-digit number less than 500, the first digit can be any of \( 1, 2, 3, 4 \) (4 choices), the second digit can be any of the remaining 9 digits (0-9 except the first digit), and the third digit can be any of the remaining 8 digits. Thus, there are: \[ 4 \times 9 \times 8 = 288 \quad \text{3-digit numbers} \] Step 4: Add the total count.
The total number of natural numbers less than 500 with no repeated digits is: \[ 9 + 81 + 288 = 378 \] Thus, the number of natural numbers less than 500 in which no two digits are repeated is \( \boxed{378} \).