How Many 4-Digit Numbers Have No 7's?

Counting and understanding the number of 4-digit numbers that do not contain the digit 7 is both a fun and practical exercise in combinatorics. This article will guide you through the process of determining the exact number of such numbers using various methods, including step-by-step calculations and elegant permutations.

Understanding the Structure of 4-Digit Numbers

A 4-digit number ranges from 1000 to 9999, and can be represented as abcd, where:

a is the thousands place, b is the hundreds place, c is the tens place, d is the units place.

Determining Valid Digits

To ensure that no 7 appears in the number, we must identify the valid digits for each place:

Thousands Place (a)

Valid digits: 1, 2, 3, 4, 5, 6, 8, 9 Total options: 8

Hundreds, Tens, and Units Places (b, c, d)

Valid digits: 0, 1, 2, 3, 4, 5, 6, 8, 9 Total options: 9

Calculating Total Combinations

Given the restriction of no 7s, we calculate the total number of 4-digit numbers by multiplying the number of options for each place:

Step-by-Step Multiplication

Choose a: 8 options (1-6, 8-9) Choose b: 9 options (0-6, 8-9) Choose c: 9 options (0-6, 8-9) Choose d: 9 options (0-6, 8-9)

Total number of 4-digit numbers with no 7s: 8 times; 9 times; 9 times; 9 8 times; 729 5832

Additional Verification Using Brute Force and J Programming Language

A thorough verification can be done using a brute force approach in the J programming language, which yields the same result:

-/..

52488

This indicates the total number of 5-digit integers without the digit 7 is 52488. However, for 4-digit numbers, we have a different count:

Combining Permutation and Combination Techniques

To form a 4-digit number without a 7, we have:

8 possible digits for the thousands position (1, 2, 3, 4, 5, 6, 8, 9), as the leading digit cannot be 0. 9 possible digits for each of the next 3 positions (0-6, 8-9).

Total number of 4-digit numbers without a 7: 8 times; 9 times; 9 times; 9 8 times; 729 5832

Conclusion

By using these techniques and verifying through various methods, we can confidently state that the total number of 4-digit numbers with no 7s is 5832. This understanding is invaluable in ensuring numbers are validated for specific digit constraints in any given task or application.