how even numbers greater than 40,000 may be formed using the digits 3,4,5,6,and 9 if each digit must be used exactly once in each number?
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To form even numbers greater than 40,000 using the digits 3, 4, 5, 6, and 9, while ensuring that each digit is used exactly once in each number, you can follow these steps:
1. Start by determining the position of the least significant digit, which should be an even digit (either 4, 6, or 9) since we want to form even numbers.
2. With the least significant digit fixed, move on to the next position (tens digit) and select one of the remaining digits (3, 5, or the other even digit not used in step 1).
3. Continue this process for the remaining positions, selecting one of the remaining unused digits each time.
4. Form numbers by permuting these selected digits in all possible ways.
Let's break it down further with an example to illustrate how to generate these numbers:
Example:
1. Start with the least significant digit. We can choose either 4, 6, or 9. Let's choose 4 as the least significant digit.
2. Move to the next position, the tens digit. There are two options left: 3 and 6. Let's choose 3 as the tens digit.
3. Move to the next position, the hundreds digit. There is only one option left: 5.
4. Move to the next position, the thousands digit. There are two options left: 6 and 9. Let's choose 6.
5. Move to the final position, the ten-thousands digit. Only one option remains: 9.
6. Now, we have the digits in order: 9, 6, 5, 3, and 4. Permute these digits to form all possible combinations: 96,534, 96,543, 95,364, etc.
In this way, you can create even numbers greater than 40,000 using the digits 3, 4, 5, 6, and 9, with each digit used exactly once in each number.