To determine how many numbers can be formed using the digits 1, 3, 5, 7, and 9 that are strictly between 5000 and 10000, and without repeating the digits, follow the steps below:
Understanding the Range: We need numbers between 5000 and 10000, which means they must be four-digit numbers starting with the digit 5, 7, or 9, as these are the only digits greater than or equal to 5 that we can use.
Counting Numbers Starting with Each Valid First Digit:
Numbers starting with 5:
If 5 is the first digit, the remaining digits (1, 3, 7, 9) can be arranged in the other three places.
There are 4 options for the second digit, 3 options for the third, and 2 options for the fourth digit.
The number of such arrangements is calculated as: \(4 \times 3 \times 2 = 24\).
Numbers starting with 7:
If 7 is the first digit, the remaining digits (1, 3, 5, 9) can be arranged in the other three places.
There are 4 options for the second digit, 3 options for the third, and 2 options for the fourth digit.
The number of such arrangements is: \(4 \times 3 \times 2 = 24\).
Numbers starting with 9:
If 9 is the first digit, the remaining digits (1, 3, 5, 7) can be arranged in the other three places.
There are 4 options for the second digit, 3 options for the third, and 2 options for the fourth digit.
The number of such arrangements is: \(4 \times 3 \times 2 = 24\).
Adding Up the Possibilities: The total number of numbers that meet the criteria is the sum of the possibilities for each starting digit:
\(24 + 24 + 24 = 72\)
Conclusion: Hence, the number of numbers strictly between 5000 and 10000 that can be formed using the digits 1, 3, 5, 7, and 9 without repetition is 72.
