2. Find the number of all natural numbers in which each subsequent digit is less than the previous one. (8 points)
Solution. The largest possible number that satisfies the condition of the problem is 9876543210. In addition, the number must be at least two digits. All other such numbers can be obtained from 9876543210 by deleting one, two, three, four, five, six, seven, or eight digits out of ten. Then the total number

$$
\begin{gathered}
1+C_{10}^{1}+C_{10}^{2}+C_{10}^{3}+C_{10}^{4}+C_{10}^{5}+C_{10}^{6}+C_{10}^{7}+C_{10}^{8}= \\
=1+C_{10}^{1}+\left(C_{10}^{2}+C_{10}^{3}+C_{10}^{4}\right) \cdot 2+C_{10}^{8}=1+10+(45+120+210) \cdot 2+252=1013
\end{gathered}
$$

Answer. 1013.