Make the greatest and the smallest:
5-digit numbers with all digits differents, given that 1 occupies the hundreds place.

Answer

To solve the problem of finding the greatest and smallest 5-digit numbers with all different digits, given that 1 occupies the hundreds place, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Fixed Digit**:
We know that the digit '1' is fixed in the hundreds place. This means our number will look like this: _ _ 1 _ _.

2. **Finding the Greatest Number**:
- To create the greatest number, we need to fill the other positions with the largest possible digits while ensuring all digits are different.
- The largest digits available (0-9) are 9, 8, 7, 6, and 5.
- We will place the largest available digit in the ten-thousands place, followed by the next largest in the thousands place, and so on.
- Therefore, we can fill the number as follows:
- Ten-thousands place: 9
- Thousands place: 8
- Hundreds place: 1 (fixed)
- Tens place: 7
- Units place: 6
- Thus, the greatest number we can form is **98176**.

3. **Finding the Smallest Number**:
- To create the smallest number, we again need to fill the other positions with the smallest possible digits while ensuring all digits are different.
- The smallest digits available (0-9) are 0, 2, 3, 4, 5, 6, 7, 8, and 9.
- However, since '1' is already in the hundreds place, we cannot use it again.
- We will place the smallest available digit in the ten-thousands place, followed by the next smallest in the thousands place, and so on.
- Therefore, we can fill the number as follows:
- Ten-thousands place: 2 (the smallest digit we can use)
- Thousands place: 0 (the next smallest digit)
- Hundreds place: 1 (fixed)
- Tens place: 3 (the next smallest digit)
- Units place: 4 (the next smallest digit)
- Thus, the smallest number we can form is **20134**.

### Final Answers:
- The greatest 5-digit number is **98176**.
- The smallest 5-digit number is **20134**.