Number System Question 7 - CAT 2023

Step-by-Step Explanation

Calculate the count of positive integers up to 1000 with non-repeating digits by grouping them by their digit count:

• Single-digit numbers (1 to 9):

All 9 numbers are valid.

Count = 9

• Two-digit numbers (10 to 99):
  • The tens place can be filled by any digit from 1 to 9 (9 choices, as 0 is invalid).
  • The units place can be filled by any digit from 0 to 9 except the one chosen for the tens place (9 remaining choices).

Count = 9 × 9 = 81

• Three-digit numbers (100 to 999):
  • The hundreds place can be filled by any digit from 1 to 9 (9 choices).
  • The tens place can be filled by any digit from 0 to 9 except the one used in the hundreds place (9 choices).
  • The units place can be filled by any digit from 0 to 9 except the two already used (8 choices).

Count = 9 × 9 × 8 = 648

• The boundary number (1000):

The number 1000 contains repeating zeros, so it is invalid.

Total valid numbers = 9 + 81 + 648 = 738

Correct Option: C