Forming Numbers Greater Than 7000 Using Specific Digits Without Repetition

In this article, we will explore how many total numbers can be formed using the digits 3, 5, 7, 8, and 9, which are greater than 7000, without repeating any digits. We will break down the problem using several methods and even provide a programming solution using the J programming language to illustrate the process in detail.

Understanding the Problem

The problem requires us to form 4-digit and potentially 5-digit numbers using the digits 3, 5, 7, 8, and 9, ensuring that no digit is repeated and the number is greater than 7000. The smallest possible number would begin with 7, and the largest with 9, comprising a range of potential combinations.

Method 1: Logical Breakdown

We start by noting that for any number greater than 7000, the first digit can only be 7, 8, or 9. Let's explore each case separately.

Case 1: First Digit is 7

After choosing 7 as the first digit, we have the remaining digits 3, 5, 8, and 9 to arrange in the subsequent positions. For each of the remaining places, we have a decreasing number of choices:

Second digit: 4 choices (3, 5, 8, 9) Third digit: 3 choices (remaining after the second choice) Fourth digit: 2 choices (remaining after the third choice)

The total number of combinations for this case is 4 x 3 x 2 24.

Case 2: First Digit is 8

Following similar logic, with the first digit 8, we have the digits 3, 5, 7, and 9 left. Again, we proceed with decreasing choices:

Second digit: 4 choices (3, 5, 7, 9) Third digit: 3 choices Fourth digit: 2 choices

The total number of combinations for this case is 4 x 3 x 2 24.

Case 3: First Digit is 9

With the first digit 9, we have the digits 3, 5, 7, and 8 left. Again, proceed with decreasing choices:

Second digit: 4 choices (3, 5, 7, 8) Third digit: 3 choices Fourth digit: 2 choices

The total number of combinations for this case is 4 x 3 x 2 24.

By summing up the combinations from all three cases, we get 24 24 24 72.

Method 2: Brute Force Solution with J Programming Language

To further validate our reasoning, we can implement a brute force solution using the J programming language. Here's the J code:

n .~7e3]10.4 perm 5{p10.5 perm 5{p.3 5 7 8 9 192

The J code yields 192, indicating that there are 192 different 4- and 5-digit numbers that meet the criteria.

Smallest and Largest Numbers

The smallest number above 7000 that can be formed is 7358, and the largest is 98753. The range of numbers includes many 4- and 5-digit combinations without repeating any digits.

Conclusion

In this article, we have detailed how to form numbers greater than 7000 from the digits 3, 5, 7, 8, and 9 without repeating any digits. By breaking down the problem logically and validating it with a programming solution, we found that there are a total of 192 such numbers. This illustrates the power of mathematical reasoning and computational methods in solving complex problems.

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