Generating 5-Digit Numbers Greater Than 50,000 Using 4, 5, 6, 7, and 8 Without Repetition

When it comes to generating 5-digit numbers greater than 50,000 using the digits 4, 5, 6, 7, and 8 without repetition, a systematic approach is key. Let's delve into the process and the underlying mathematics to explore how many such numbers can be formed.

Understanding the Constraint

The primary constraint is that the number must be a 5-digit number greater than 50,000. This means that each such number will have all five digits included, and the first digit must be one of 5, 6, 7, or 8 to ensure the number is greater than 50,000.

Steps to Form 5-Digit Numbers

To solve the problem systematically, we can break it down into two main steps:

Determining the First Digit

The first digit of the number must be one of 5, 6, 7, or 8. There are 4 choices for the first digit. Once the first digit is chosen, it leaves 4 digits to be arranged in the remaining 4 positions.

Arranging the Remaining Digits

After choosing the first digit, the remaining 4 digits can be arranged in any order. The number of permutations of 4 digits is given by 4!, which equals 24.

Calculating the Total Number of Permutations

To find the total number of 5-digit numbers greater than 50,000, we multiply the number of choices for the first digit by the number of permutations of the remaining 4 digits.

Mathematically, this can be expressed as:

Total permutations Number of choices for the first digit × Number of permutations of remaining 4 digits

Total permutations 4 × 4!

Total permutations 4 × 24 96

Therefore, there are 96 possible 5-digit numbers greater than 50,000 that can be formed using the digits 4, 5, 6, 7, and 8 without repetition.

Example and Verification

To better illustrate the process, let's consider a few examples.

If the first digit is 5, the remaining digits (4, 6, 7, 8) can be arranged as follows:

54678 54687 54768 54786 54867 54876 56478 56487 ... 58764 58746 58674

Similarly, if the first digit is 6, 7, or 8, the same logic applies, and each case will yield 24 possible numbers. Thus, multiplying by the 4 choices for the first digit, we get a total of 96 numbers.

Conclusion

By understanding the constraint and using combinatorial techniques, we can systematically determine the number of 5-digit numbers greater than 50,000 that can be formed with the digits 4, 5, 6, 7, and 8 without repetition. The solution involves calculating permutations with a constraint, which is a fundamental concept in combinatorics and number theory.

References

Combinatorics and Permutations: Wikipedia () Permutations with Constraints: Discrete Mathematics (_Mathematics/9:_Fundamental_Counting_Principles/9.6:_Permutations_With_Constraints)