There are 4536 numbers in the range 1000 to 9999 that have no repeated digits.
Understanding Four-Digit Numbers with Unique Digits
Numbers in the range 1000 to 9999 are precisely the four-digit integers. To determine how many of these numbers have no repeated digits, we need to count the arrangements of four unique digits, where the first digit cannot be zero. This is a classic combinatorial problem that involves selecting digits for each position without replacement.
Let's break down the process for constructing such a number:
Step-by-Step Digit Selection
We need to select four distinct digits for the thousands, hundreds, tens, and units places.
-
Thousands Place (First Digit):
- Since it's a four-digit number, the first digit cannot be 0.
- Therefore, there are 9 possible choices (1, 2, 3, 4, 5, 6, 7, 8, 9).
-
Hundreds Place (Second Digit):
- Now, we can use 0. However, the digit must be different from the one chosen for the thousands place.
- Since one digit has already been used, there are 10 (total digits 0-9) - 1 (digit used for thousands place) = 9 possible choices remaining.
-
Tens Place (Third Digit):
- The digit for this place must be different from the two digits already chosen for the thousands and hundreds places.
- Therefore, there are 10 (total digits 0-9) - 2 (digits used) = 8 possible choices remaining.
-
Units Place (Fourth Digit):
- Finally, the digit for this place must be different from the three digits already chosen for the thousands, hundreds, and tens places.
- This leaves 10 (total digits 0-9) - 3 (digits used) = 7 possible choices remaining.
Calculation of Total Numbers
To find the total number of such integers, we multiply the number of choices for each position:
Total Numbers = (Choices for Thousands Place) × (Choices for Hundreds Place) × (Choices for Tens Place) × (Choices for Units Place)
Total Numbers = 9 × 9 × 8 × 7
Let's perform the multiplication:
- 9 × 9 = 81
- 8 × 7 = 56
- 81 × 56 = 4536
| Digit Place | Number of Choices | Explanation |
|---|---|---|
| Thousands | 9 | (1-9; cannot be 0) |
| Hundreds | 9 | (0-9, excluding the digit used for thousands place) |
| Tens | 8 | (0-9, excluding the two digits used for thousands and hundreds) |
| Units | 7 | (0-9, excluding the three digits used previously) |
| Total Numbers | 4536 | (9 × 9 × 8 × 7) |
Therefore, there are 4536 numbers between 1000 and 9999 (inclusive) that have no repeated digits.
