Determine the number of all positive ten-digit integers with the following properties:

- The number contains each of the digits 0, 1, 2, ..., 8, and 9 exactly once.
- Each digit, except for the 9, has a neighboring digit that is greater than it.

(Note. For example, in the number 1230, the digits 1 and 3 are the neighboring digits of 2, and 2 and 0 are the neighboring digits of 3. The digits 1 and 0 have only one neighboring digit.)

(Karl Czakler)

Answer. There are 256 numbers with the required properties.
Let $A$ be a ten-digit number with this property. The place value of the digit 9 is $10^{k}$ with $k \in\{0,1,2, \ldots, 9\}$. All digits of the number $A$ that are to the left of 9 must be arranged in ascending order, while all digits to the right of 9 must be arranged in descending order. It follows immediately that the digit 0 can only occur at the unit's place and therefore the digit 9 cannot be at the unit's place, i.e., it must be $k>0$.

We therefore have:

- For $k=9$ there is only one number, namely 9876543210.
- For $k=8$ there are $\left(\begin{array}{l}8 \\ 1\end{array}\right)=8$ numbers, because one of the remaining 8 digits must be chosen to be placed before the digit 9. An example is the number 3987654210.
- For $k=7$ there are $\left(\begin{array}{l}8 \\ 2\end{array}\right)=28$ numbers, because there are $\left(\begin{array}{l}8 \\ 2\end{array}\right)$ ways to choose 2 digits from the remaining 8 digits to be placed before the 9. Their order is already determined by this choice, as they must be arranged in ascending order. An example is the number 1498765320.
- For $2 \leq k \leq 6$ there are, by the same reasoning, $\left(\begin{array}{c}8 \\ 9-k\end{array}\right)$ possibilities (numbers) for each $k$.
- For $k=1$ there is again only one $\left(1=\left(\begin{array}{l}8 \\ 8\end{array}\right)\right)$ number, namely 1234567890.

In total, there are therefore

$$
\left(\begin{array}{l}
8 \\
0
\end{array}\right)+\left(\begin{array}{l}
8 \\
1
\end{array}\right)+\left(\begin{array}{l}
8 \\
2
\end{array}\right)+\cdots+\left(\begin{array}{l}
8 \\
8
\end{array}\right)=2^{8}=256
$$

numbers with the required property.

(Karl Czakler)