Counting 5-Digit Numbers: A Comprehensive Analysis

In this article, we will explore how to calculate the total number of 5-digit numbers between two given points, namely 9999 and 99999. We will present different methods and use various programming languages to arrive at the solution. By the end of this article, you will have a solid understanding of the techniques involved in count such numbers.

Mathematical Approach

To start, let's analyze the problem using pure mathematics. A 5-digit number lies between 9999 and 99999. The first digit can be any number from 9 to 9, giving us 9 possibilities. The remaining four digits can be any number from 0 to 9, each with 10 possibilities.

The first digit has nine possibilities (excluding 0). Each of the remaining four digits has 10 possibilities.

The total number of 5-digit numbers can be calculated as follows:

Number of 5-digit numbers 9104 90000

Brute Force Approach Using J Programming Language

The J programming language offers a simple and efficient way to calculate the number of 5-digit numbers within a specified range. Let's use a brute force approach to count the numbers from 10000 to 99999.

10000 to 9999990000

Thus, we can conclude that there are 90000 5-digit integers within the range from 9999 to 99999.

Arithmetic Progression Method

To further validate our solution, let's consider using the concept of arithmetic progression (AP). The first 5-digit number is 10000, and the last 5-digit number just below 99999 is 99998.

In an AP, the first term (a) is 10000, the common difference (d) is 1, and the last term (Tn) is 99998.

The last term Tn is 99998. The common difference d is 1. The first term a is 10000.

The formula to find the number of terms (n) in an AP is given by:

n - 1d Tn - a

Solving the equation:

n - 1 99998 - 10000 899991

Hence, n 89999.

Therefore, the number of 5-digit numbers from 9999 to 99999 is 89999.

Conclusion

In conclusion, there are 89999 five-digit numbers between 9999 and 99999. Whether you use mathematical reasoning, brute force programming, or the concept of arithmetic progression, the result is consistent and reliable.

For more such number theory problems, and to enhance your understanding of these concepts, consider exploring similar problems or consulting additional resources.

Key Points:

99999 - 9999 - 1 90000 (brute force approach) 10000 to 99998 (arithmetic progression method) 9104 90000 (mathematical approach)