## Task $2 / 78$

Determine the number $A(n)$ of at least two-digit natural numbers (in decimal notation) that have the following property:

Each digit with a higher place value is smaller than each digit with a lower place value.
The largest number with the required property is obviously the number 123456 789. From this, one obtains all relevant numbers with $9-i$ digits by striking out $i$ arbitrary digits (where $i \in N, 0 \leq i \leq 7$). This is possible in $\binom{9}{i}$ different ways. The desired number $A(n)$ is thus the sum of the binomial coefficients $\binom{9}{i}$ with $0 \leq i \leq 7$:

$$
A(n)=\sum_{i=0}^{7}\binom{9}{i}=1+9+36+84+126+126+84+36=502
$$