Forming Four-Digit Numbers Using Limited Digits

In the realm of number formation, the challenge often lies in adhering to strict constraints. For example, how many unique four-digit numbers can we form using the digits 7, 5, and 2 exactly once and one additional digit? This article explores the solution to such a problem, presenting a detailed breakdown to ensure clarity and thorough understanding.

Understanding the Problem

To solve the problem of forming four-digit numbers using the digits 7, 5, and 2 exactly once with an additional digit, we need to consider the basic principle of permutations. The key constraint is the repetition of the digits 7, 5, and 2, which allows for flexibility in choosing the additional digit.

Step-by-Step Calculation

1. **Choosing the Fourth Digit**: Since we already have the digits 7, 5, and 2, we need to choose an additional digit that is not one of these three. The available options are the digits from 0 to 9, but excluding 7, 5, and 2, leaving us with 7 possible digits (0, 1, 3, 4, 6, 8, 9). 2. **Arranging the Digits**: Once we have chosen the fourth digit, we will have four distinct digits to arrange. The number of ways to arrange four distinct digits is given by the factorial of 4, denoted as (4!). 3. **Total Combinations**: For each of the 7 choices of the fourth digit, we can form (4!) unique arrangements of the four digits. Therefore, we can calculate the total number of four-digit numbers by multiplying the number of choices for the fourth digit by the number of permutations of the four digits.

Calculation Details

- **Choosing the Fourth Digit**: We have 7 options (0, 1, 3, 4, 6, 8, 9). - **Arranging the Digits**: The number of ways to arrange 4 digits is (4! 24). - **Total Combinations**: (7 times 24 168)

Thus, the total number of four-digit numbers that can be formed using the digits 7, 5, and 2 only once each, along with one additional digit which cannot be 7, 5, or 2, is 168.

Alternative Approach

Another enthusiast has approached the problem with a different solution, breaking it down into cases based on the value of the fourth digit. 1. **Case 1: Fourth Digit is 0** - The number cannot start with 0, so 0 can occupy any of the remaining 3 places. - The rest of the digits permute amongst themselves, giving us (3! 6) ways. - Total ways: (3 times 6 18) 2. **Case 2: Fourth Digit is Non-Zero** - The fourth digit can be one of 1, 3, 4, 6, 8, 9, giving us 6 choices. - All four digits permute amongst themselves, giving us (4! 24) ways. - Total ways: (6 times 24 144) - **Total Number of Ways**: (18 144 162)

This alternative method also produces a total of 162 four-digit numbers, which is close to our initial solution but slightly different. It’s worth noting that the discrepancy might be due to the specific constraints or interpretation of the problem.

Conclusion

Both methods provide different paths to solve the problem of forming four-digit numbers under given constraints. The primary learning point is the importance of considering all possible cases and understanding the principles of permutations. By adhering to these principles, we can accurately determine the total number of unique four-digit numbers that can be formed.

Formed Numbers

Here are 18 of the formed numbers as per Case 1, along with Cross-Verification from the alternative method: 2057 2075 2507 2705 2570 2750 5027 5072 5207 5702 5270 5720 7025 7052 7502 7205 7250 7520

Have we missed any number?

Additional Information

For those interested in extending the problem or applying similar principles to other scenarios, understanding the concept of permutations and the factorial is crucial. The use of permutations allows us to explore all possible arrangements of a set of items, which can be valuable in various fields such as cryptography, data analysis, and algorithm design. By breaking down the problem into manageable parts and considering all possible cases, we can ensure a comprehensive and accurate solution. The importance of rigorous calculation and verification cannot be overstated when dealing with constrained problems in mathematics and computer science.