Counting Odd Three-Digit Numbers

The question of counting odd three-digit numbers is a common exploration in the realm of number theory. This article will delve into the logic and steps used to determine the total count of such numbers, providing a comprehensive understanding.

Introduction

Odd three-digit numbers are numbers between 100 and 999 that do not end in an even digit. In this context, the odd digits available are 1, 3, 5, 7, and 9. Understanding how to count these numbers involves analyzing the constraints on each digit and applying the fundamental counting principle.

Calculating the Total

All Digits Odd

To determine the total number of odd three-digit numbers where all digits are odd, we can consider the hundreds, tens, and units places.

The hundreds place can be any of the 5 odd digits (1, 3, 5, 7, 9). For the tens and units places, we also have 5 choices. Therefore, the total number of such numbers is calculated as follows:

[ 5 times 5 times 5 125 ]

First Digit Not Zero

For the more specific scenario where the first digit (hundreds place) is non-zero, and all three digits are odd, we follow a similar calculation. This is because we have 5 options (1, 3, 5, 7, 9) for the hundreds place, and 5 options (0, 2, 4, 6, 8) for the tens and units places (but they are all odd in this specific case).

[ 5 times 5 times 5 125 ]

Repetition and Negative Numbers

If we consider allowing repetition of digits (each digit can be 1, 3, 5, 7, 9), the calculation remains the same as previously discussed.

However, if we restrict repetition, we would have to consider permutations. For the first digit, we have 5 choices, the second digit (tens place) also has 5 choices, but the third digit (units place) depends on the choices made for the first two digits. This scenario is less common but can be calculated as follows:

[ 5 times 5 times 4 100 ]

Note that the 4 choices for the third digit account for the fact that it cannot be the same as the first digit in this reduced set.

Additional Considerations

When considering the total numbers between 100 and 999, excluding numbers with leading zeros (011, 033, etc.), there are 899 valid three-digit numbers. Since half of these numbers are odd, the count of odd three-digit numbers is:

[ frac{899}{2} 450 ]

Conclusion

In summary, the total number of odd three-digit numbers where all digits are odd can be determined using the fundamental counting principle, with 5 choices for each odd digit position. For 125, this is the definitive count if repetition is allowed across all digits.

The results underscore the importance of understanding the constraints and applying basic arithmetic principles to derive the correct answer.