The exact number of odd integers from 1000 through 9999 that have distinct digits is 2240.
Unveiling the Count: Odd Integers with Distinct Digits
Determining the quantity of odd integers within a specific range that also feature distinct digits involves a methodical application of counting principles. For numbers between 1000 and 9999, we are dealing with four-digit integers. The constraints are that each digit must be unique, and the number itself must be odd.
Breakdown of the Calculation Process
To ensure all conditions are met, we consider the choices for each digit place, starting with the most restrictive condition – the units digit determining if the number is odd.
Let the four-digit number be represented as $ABCD$, where:
- $A$ is the thousands digit.
- $B$ is the hundreds digit.
- $C$ is the tens digit.
- $D$ is the units digit.
Here's how we calculate the possibilities:
-
Units Digit ($D$): For a number to be odd, its units digit must be odd.
- Choices: {1, 3, 5, 7, 9}
- Number of options: 5
-
Thousands Digit ($A$): This digit has two main restrictions:
- It cannot be 0 (as it's a four-digit number from 1000).
- It must be distinct from the units digit ($D$).
- Since one digit ($D$) has already been chosen from the 10 available digits (0-9), 9 digits remain. From these 9, we must exclude 0.
- Number of options: 8 (10 total digits - 1 (for D) - 1 (for 0) = 8)
-
Hundreds Digit ($B$): This digit must be distinct from the thousands digit ($A$) and the units digit ($D$).
- Two digits ($A$ and $D$) have already been chosen.
- Number of options: 8 (10 total digits - 2 used = 8)
-
Tens Digit ($C$): This digit must be distinct from the thousands ($A$), hundreds ($B$), and units ($D$) digits.
- Three digits ($A$, $B$, and $D$) have already been chosen.
- Number of options: 7 (10 total digits - 3 used = 7)
Summary of Digit Choices
To visualize the choices, consider the following table:
| Digit Place | Conditions | Number of Choices |
|---|---|---|
| Units (D) | Must be odd | 5 |
| Thousands (A) | Cannot be 0, must be distinct from D | 8 |
| Hundreds (B) | Must be distinct from A and D | 8 |
| Tens (C) | Must be distinct from A, D, and B | 7 |
Final Calculation
To find the total number of such integers, we multiply the number of choices for each digit place:
Total = (Choices for D) × (Choices for A) × (Choices for B) × (Choices for C)
Total = 5 × 8 × 8 × 7
Total = 40 × 56
Total = 2240
This calculation confirms the figure provided by educational resources. As stated in the reference from homework.study.com, there are indeed 2240 odd numbers between 1000 and 9999 that have distinct digits. This detailed approach ensures accuracy and clarity in arriving at the precise count.
