To solve the problem, we need to find the number of nine-digit numbers that contain each digit from 1 to 9 exactly once, with the condition that the digits 1 to 5 are in their natural order, while the digits 1 to 6 are not in their natural order.
### Step-by-Step Solution:
1. **Understanding the Natural Order Requirement**:
- The digits 1, 2, 3, 4, and 5 must appear in the order 1 < 2 < 3 < 4 < 5. This means that wherever these digits appear in the nine-digit number, they must maintain this sequence.
2. **Choosing Positions for Digits 1 to 5**:
- We need to select 5 positions out of the 9 available positions for the digits 1, 2, 3, 4, and 5. The number of ways to choose 5 positions from 9 can be calculated using the combination formula:
\[
\binom{9}{5} = \frac{9!}{5!(9-5)!} = \frac{9!}{5!4!} = 126
\]
3. **Arranging Digits 1 to 5**:
- Once we have chosen the 5 positions, the digits 1, 2, 3, 4, and 5 will automatically occupy these positions in their natural order. Thus, there is only 1 way to arrange these digits in the chosen positions.
4. **Arranging Digits 6, 7, 8, and 9**:
- The remaining 4 positions will be filled with the digits 6, 7, 8, and 9. These digits can be arranged in any order. The number of ways to arrange 4 digits is given by:
\[
4! = 24
\]
5. **Condition for Digits 1 to 6**:
- The digits 1 to 6 must not be in natural order. The only arrangement that keeps them in natural order is 1, 2, 3, 4, 5, 6. We need to exclude this arrangement.
- Since we already have the digits 1 to 5 in order, we can place the digit 6 in any of the remaining positions (which are not occupied by 1 to 5). This means we can place 6 in any of the 4 remaining positions.
6. **Calculating the Total Valid Arrangements**:
- For each arrangement of 1 to 5, we can place 6 in any of the remaining 4 positions, and then arrange 7, 8, and 9 in the remaining 3 positions. Thus, the total number of valid arrangements is:
\[
\text{Total arrangements} = \binom{9}{5} \times 1 \times 4! = 126 \times 24 = 3024
\]
- However, we need to subtract the arrangement where 1, 2, 3, 4, 5, and 6 are in natural order (which is just 1 arrangement). Therefore:
\[
\text{Final count} = 3024 - 1 = 3023
\]
### Final Answer:
The total number of such nine-digit numbers is **3023**.
