Forming Unique 3-Digit Numbers Using Digits 1, 2, 3, and 4

When working with digits, understanding how to form unique numbers is a fundamental concept, whether through permutations with or without repetition. In this article, we will delve into the problem of forming 3-digit numbers using the digits 1, 2, 3, and 4. We will explore both scenarios: where repetition of digits is allowed and where it is not.

Understanding the Problem

Given a set of four digits (1, 2, 3, 4), we aim to determine how many unique 3-digit numbers can be formed. The core principles to consider are the availability of digits and the number of digits to be used. Each digit can be reused in the case of repetition, but cannot be reused if repetition is not allowed.

Calculations with Repetition Allowed

When repetition of digits is allowed, each of the three positions (hundreds, tens, and units) in the number can contain any of the four digits. Therefore, for each position, there are four possible choices.

Hundreds place: 4 choices (1, 2, 3, or 4)

Tens place: 4 choices (1, 2, 3, or 4)

Units place: 4 choices (1, 2, 3, or 4)

Thus, the total number of 3-digit numbers that can be formed is calculated as follows:

Total 4 * 4 * 4 43 64

Calculations with Repetition Not Allowed

When repetition is not allowed, we must choose each digit from the available set of four without reuse.

First digit: 4 choices

Second digit: 3 remaining choices

Third digit: 2 remaining choices

Therefore, the total number of unique 3-digit numbers that can be formed is:

Total 4 * 3 * 2 24

Visual Representation and Step-by-Step Solution

To form a 3-digit number, we can represent this visually in three spaces:

______ ______ ______

1. For the first space (hundreds place), we have 4 possible choices.

2. For the second space (tens place), we have 3 remaining choices (since one digit is used in the first space).

3. For the third space (units place), we have 2 remaining choices (since two digits are used in the first and second spaces).

Thus, the total number of unique 3-digit numbers without repetition is:

Total 4 * 3 * 2 24

Conclusion

In summary, when using the digits 1, 2, 3, and 4, we can form 64 unique 3-digit numbers if repetition is allowed, and 24 unique 3-digit numbers if repetition is not allowed.

By understanding these calculations, you can effectively approach similar problems involving permutations and combinations in various mathematical and real-world scenarios.