4. In the place of Donje Zuče, every phone number has five digits arranged in strictly increasing or strictly decreasing order, and the first digit is not 0. How many phone numbers can exist in that place?
4. Let's first count how many such numbers can have digits in descending order. Each such number is uniquely determined by the selection of five (different) digits that compose it (once the digits are chosen, the number is obtained by sorting them in descending order), so since we have 10 digits available, there are $\binom{10}{5}$ such numbers. Now let's count the numbers whose digits are in ascending order. Similarly to the previous case, each such number is uniquely determined by the selection of the digits that compose it; this time, however, we have only 9 digits available from which to choose 5, because the digit 0 cannot be among the chosen digits (if it were, it would have to go at the beginning of the telephone number due to the ascending order, which is prohibited by the problem's condition). Thus, in this case, we have $\left.\begin{array}{l}9 \\ 5\end{array}\right)$ numbers. The total result is: $\binom{10}{5}+\binom{9}{5}=252+126=378$ telephone numbers.