How Many Three-Digit Numbers Can Be Formed and How Many Are Odd?

Introduction: In the world of mathematics, understanding the permutations and constraints of forming numbers is fundamental. This article explores the detailed process of forming three-digit numbers from a set of digits without repetition, and how many of these numbers are odd.

Understanding the Problem

The problem at hand involves forming three-digit numbers using the digits 0, 1, 2, 3, 4, 5, and 6, with the condition that each digit can only be used once. We will break down the problem into two parts: the total number of three-digit numbers that can be formed, and the number of those that are odd.

Total Three-Digit Numbers

A three-digit number can be represented as XYZ, where X is the hundreds place, Y is the tens place, and Z is the units place. However, X cannot be 0, as it would then fall outside the three-digit number range.

Breaking Down the Choices

Choosing X (Hundreds Place): Since X cannot be 0, there are 6 possible choices (1, 2, 3, 4, 5, 6). Choosing Y (Tens Place): Y can be any of the remaining 7 digits (0, 1, 2, 3, 4, 5, 6), minus the one already used for X. Choosing Z (Units Place): Z can be any of the remaining 6 digits (after choosing X and Y).

Thus, the total number of three-digit numbers can be calculated as follows:

text{Total} 6 times; 6 times; 5 180

Three-Digit Odd Numbers

For a number to be odd, the last digit (Z) must be one of the odd digits: {1, 3, 5}. Therefore, there are 3 choices for Z.

Breaking Down the Choices Again

Choosing Z (Units Place): Z can be one of the three odd digits (1, 3, 5). Choosing X (Hundreds Place): X can be any digit except 0 and cannot be the same as Z. Therefore, there are 5 choices left (since one digit is already used for Z). Choosing Y (Tens Place): Y can be any of the remaining 5 digits (after choosing X and Z).

The total number of three-digit odd numbers can be calculated as follows:

text{Odd numbers} 3 times; 5 times; 5 75

Summary

Total three-digit numbers: 180 Total three-digit odd numbers: 75

Conclusion

Understanding the constraints and applying the principles of permutations is crucial in solving such problems. By breaking down the choices systematically, we can easily determine the total number of three-digit numbers that can be formed from the given digits and how many of these numbers are odd.