10.2. Find the number of all five-digit numbers $\overline{a b c d e}$, all digits of which are distinct and $ad>e$.
Answer: 1134.

Solution. For the correct notation of a number satisfying the condition of the problem, one needs to arbitrarily select a quintet of different digits from 10 possible ones, and then arrange two of them to the left of the maximum in ascending order and the two remaining to the right of the maximum in descending order. There are two different cases possible.

1) The selected quintet does not contain zero. Then it can be chosen in $C_{9}^{5}$ ways, then choose two non-maximum and non-zero digits from them in $C_{4}^{2}$ ways, and arrange them to the left of the largest in ascending order in a unique way, and write the two remaining to the right of the maximum in descending order. In total, we have $C_{9}^{5} \cdot C_{4}^{2}=126 \cdot 6=756$ such numbers.
2) The selected quintet contains zero. Then its non-zero digits can be chosen in $C_{9}^{4}$ ways, then choose two non-maximum digits from them in $C_{4}^{2}$ ways, and then arrange them to the left of the largest in ascending order in a unique way, and write one remaining and zero to the right of the maximum in descending order. In total, we have $C_{9}^{4} \cdot C_{3}^{2}=126 \cdot 3=378$ such numbers.

In total, we get $756+378=1134$ numbers.

Grading criteria. The idea of separately considering the cases when there is a 0 among the digits of the number and when there is not: 1 point. Correct consideration of each case: 3 points.