Forming Odd Numbers Greater Than 70000 Using Specific Digits
When dealing with the formation of numbers, ensuring that the number fits specific criteria such as being odd and greater than 70000 can become quite a challenge. This article focuses on the steps and logic behind forming five-digit and six-digit odd numbers using the digits 0, 1, 2, 7, 8, and 9 without repetition. The process involves deploying the principles of permutations and carefully analyzing each digit's role to meet the overall requirements.
Understanding the Criteria
To form an odd number greater than 70000, we need to ensure the number meets two primary conditions:
The number should be odd, meaning the last digit must be one of the odd digits (1, 7, or 9). The number must be greater than 70000, meaning the first digit must be one of 7, 8, or 9.Step-by-Step Formation Process
The process involves starting with the most restrictive requirement (the last digit) and moving to the next most restrictive one (the first digit), finally completing the number with the remaining digits.
1. Last Digit - Must be Odd
The last digit has to be one of the three odd digits: 1, 7, or 9.
Error
Let's break it down:
If the last digit is 1: 5-digit numbers: The first digit can be 7, 8, or 9 (3 choices). The remaining three digits can be any of the remaining digits (4 choices for the second digit, 3 for the third, and 2 for the fourth). Hence, the number of 5-digit numbers ending with 1 is: 3 x 4 x 3 x 2 72. 6-digit numbers: The first digit can be 2, 8, or 9 (3 choices). The remaining four digits can be any of the remaining digits (4 choices for the second digit, 3 for the third, 2 for the fourth, and 1 for the fifth). Hence, the number of 6-digit numbers ending with 1 is: 3 x 4 x 3 x 2 x 1 96.Total: 72 96 168
2. Last Digit - Must be 7 or 9
If the last digit is 7 or 9:
5-digit numbers: The first digit can be 8 or 9 (2 choices). The remaining three digits can be any of the remaining digits (4 choices for the second digit, 3 for the third, and 2 for the fourth).Hence, the number of 5-digit numbers ending with 7 or 9 is: 2 x 4 x 3 x 2 48.
Total for 5-digit numbers ending with 7 or 9: 48 48 96
Total for 6-digit numbers ending with 7 or 9: 4 x 4 x 3 x 2 x 1 96.
Total: 96 96 192
3. Summing Up All Possibilities
To find the total number of five-digit and six-digit odd numbers greater than 70000:
Total: 168 192 360.
Conclusion
This breakdown illustrates the logical process and precise calculations required to form the desired numbers. By systematically addressing the most restrictive requirements followed by the next, we can ensure that all possible numbers are accounted for. This method not only solves the current problem but also provides a framework for tackling similar problems in the future.