Calculating Four-Digit Combinations with Digit Repetition: 1234
In this article, we will explore how many four-digit combinations can be made using the digits 1, 2, 3, and 4, where repetition of the digits is allowed. This problem falls under the category of combinations with repetition, a fundamental concept in combinatorics. We will delve into the mathematical formula and provide a step-by-step solution to calculate the total number of possible combinations.
Understanding the Problem
The question at hand is how many different four-digit numbers can be formed using the digits 1, 2, 3, and 4, where the digits can be repeated. This means that each digit of the four-digit number can independently be any of the four digits, leading to a variety of possible combinations.
Mathematical Approach
To solve this problem, we use the formula for combinations with repetition. The formula is given by:
Total combinations nr
Where:
n is the number of options (in this case, the number of available digits). r is the number of positions (in this case, the number of digits in the combination).In our specific scenario:
n 4 (digits 1, 2, 3, 4) r 4 (four-digit combination)Therefore, we need to calculate:
Total combinations 44
Performing the calculation:
44 256
Hence, there are 256 different four-digit combinations that can be made using the digits 1, 2, 3, and 4 with repetition allowed.
Breaking Down the Calculation
Let's break down the process step by step:
For the first digit, we have 4 possible choices: 1, 2, 3, or 4. Since digits can be repeated, we have 4 options for the second digit as well. The same applies to the third and fourth digits, giving us 4 options each. Thus, the total number of possible combinations is calculated as:4 x 4 x 4 x 4 256
This calculation can also be represented as 44 256, which confirms our initial formula and solution.
Conclusion
In conclusion, the total number of four-digit combinations that can be generated using the digits 1, 2, 3, and 4 with repetition is 256. This problem exemplifies the application of the formula for combinations with repetition in a real-world context. Understanding and correctly applying such formulas can be immensely useful in various fields, including computer science, cryptography, and data analysis.