There are 648 numbers between 100 and 1000 in which all the digits are distinct.
Understanding Numbers Between 100 and 1000
Numbers between 100 and 1000 are essentially three-digit numbers, ranging from 101 to 999. The task is to find how many of these three-digit numbers have unique (distinct) digits in all three positions (hundreds, tens, and units).
Calculating Numbers with Distinct Digits
To determine the number of such integers, we consider the choices for each digit place:
-
Hundreds Digit (First Digit):
- Since the number must be between 100 and 1000, it's a three-digit number. Therefore, the hundreds digit cannot be 0.
- Possible digits are 1, 2, 3, 4, 5, 6, 7, 8, 9.
- Number of choices: 9
-
Tens Digit (Second Digit):
- The tens digit can be any digit from 0 to 9, but it must be different from the hundreds digit (to ensure distinctness).
- Since one digit has already been used for the hundreds place, and there are 10 total digits (0-9), there are 9 remaining unique digits available for the tens place.
- Number of choices: 9
-
Units Digit (Third Digit):
- The units digit can be any digit from 0 to 9, but it must be different from both the hundreds digit and the tens digit (to maintain distinctness across all three positions).
- Since two distinct digits have already been used for the hundreds and tens places, there are 8 remaining unique digits available for the units place.
- Number of choices: 8
Summarizing the Choices
| Digit Place | Available Digits (0-9) | Restrictions | Number of Choices |
|---|---|---|---|
| Hundreds | 1, 2, 3, 4, 5, 6, 7, 8, 9 | Cannot be 0 | 9 |
| Tens | 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 | Must be different from the Hundreds digit | 9 |
| Units | 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 | Must be different from Hundreds and Tens | 8 |
Final Calculation
To find the total number of such numbers, we multiply the number of choices for each digit place:
Total numbers = (Choices for Hundreds Digit) × (Choices for Tens Digit) × (Choices for Units Digit)
Total numbers = 9 × 9 × 8 = 648
As confirmed by the reference from Toppr.com, "Hence, the total number of required numbers =9×9×8=648."
This calculation yields the exact answer for numbers between 100 and 1000 (i.e., three-digit numbers) where all digits are distinct.
