5-3. How many two-digit numbers exist where at least one of the digits is smaller than the corresponding digit in the number $35?$

For example, the numbers 17 and 21 are valid, while the numbers 36 and 48 are not.
Answer: 55.

Solution. First, let's find the number of two-digit numbers that do not meet the condition. In the units place, any digit from 5 to 9 can stand, and in the tens place, from 3 to 9. The total number of numbers that do not suit us will be exactly $7 \cdot 5=35$. Now we can count the number of two-digit numbers that meet the condition: we need to subtract the number of unsuitable numbers from their total number, which is 90: $90-35=55$.