Generating 5-Digit Numbers Using Specific Digits

Imagine you are tasked with generating all possible 5-digit numbers using the digits 1, 3, 5, 7, 8, and 9. The challenge is to form numbers where the first digit is 7, and one of the other digits must be 1. Additionally, no digit can appear more than once in any number. How many such numbers can be formed?

Understanding the Problem

The problem can be broken down into manageable steps. First, we place the digit 7 as the first digit. This leaves us with 5 remaining digits (1, 3, 5, 8, 9), from which we need to form a 4-digit number with the constraint that one of the digits must be 1.

Step-by-Step Solution

Step 1: Fix the First Digit and Place the Digit 1

With the digit 7 fixed as the first digit, we have the remaining 5 digits (1, 3, 5, 8, 9) to form a 4-digit number. The digit 1 must be included in these 4-digit numbers. The digit 1 can occupy any one of the 4 remaining positions in the 4-digit number. This gives us 4 options for the position of the digit 1.

Step 2: Arrange the Remaining Digits

Once the position of the digit 1 is fixed, we have 4 remaining digits (3, 5, 8, 9) to arrange in the 3 other positions. The number of ways to arrange 4 digits into 3 positions is given by the permutation formula 4P3, which is calculated as 4! / (4-3)!. This evaluates to 4! 4 × 3 × 2 × 1 24.

Combining the Steps

Since we have 4 options for placing the digit 1 and 24 ways to arrange the remaining 3 digits, the total number of possible 5-digit numbers is the product of these two values:

Total number of numbers 4 × 24 96.

Brute Force Solution Using J Programming Language

To verify this solution, we can use the J programming language to generate all possible 5-digit numbers and count those that meet the criteria. The J code provided in the initial problem statement can be used as follows:

n.5 perm 6{1 3 5 7 8 9720 5/m../96

The answer is 96, confirming that 96 five-digit integers meet all the stipulated criteria.

To list all these numbers, we can use the following J code snippet:

4 24/:~71358 71359 71385 71389 71395 71398 71538 71539 71583 71589 71593 71598 71835 71839 71853 71859 71893 71895 71935 71938 71953 71958 71983 71985 73158 73159 73185 73189 73195 73198 73518 73519 73581 73591 73815 73819 73851 73891 73915 73918 73951 73981 75138 75139 75183 75189 75193 75198 75318 75319 75381 75391 75813 75819 75831 75891 75913 75918 75931 75981 78135 78139 78153 78159 78193 78195 78315 78319 78351 78391 78513 78519 78531 78591 78913 78915 78931 78951

This list confirms the solution by showing all 96 valid 5-digit numbers.

Conclusion

By following a structured approach, we can generate all possible 5-digit numbers using the digits 1, 3, 5, 7, 8, and 9, with the first digit as 7, one of the other digits being 1, and no digit appearing more than once. The total number of such numbers is 96.