Question

From 100 to 1000 how many 3 digit numbers are there with all digits in its distinct?

• A. 900
• B. 899
• C. 648
• D. 675

Answer 1

Answer:

C. 648

Objective: Counting 3-digit numbers with distinct digits.

• Understanding the Problem: We need to find the count of numbers between 100 and 1000 (inclusive of 100, exclusive of 1000) where all three digits are different. These are essentially 3-digit numbers.

• Constraints:

  • The number must be a 3-digit number, meaning it ranges from 100 to 999.

  • All digits in the number must be unique (no repetition).

• Methodology: Using Permutations and Combinations Principles.

• Step 1: Choosing the Hundreds Digit.

  • The hundreds digit cannot be 0 (otherwise, it would be a 2-digit number).

  • So, there are 9 options for the hundreds digit (1, 2, 3, 4, 5, 6, 7, 8, 9).

• Step 2: Choosing the Tens Digit.

  • The tens digit can be any digit from 0 to 9.

  • However, it cannot be the same as the hundreds digit.

  • Thus, there are 10 (total digits) - 1 (hundreds digit) = 9 options for the tens digit.

• Step 3: Choosing the Units Digit.

  • The units digit can be any digit from 0 to 9.

  • It cannot be the same as the hundreds digit OR the tens digit.

  • Thus, there are 10 (total digits) - 2 (hundreds and tens digits) = 8 options for the units digit.

• Total Count Calculation:

  • To find the total number of 3-digit numbers with distinct digits, we multiply the number of options for each position:

  • Total numbers = (Options for hundreds digit) × (Options for tens digit) × (Options for units digit)

  • Total numbers = 9 × 9 × 8 = 648