We will call a three-digit natural number "even-loving" if it contains two even digits and the digit 1. We will call a three-digit natural number "odd-loving" if it contains two odd digits and the digit 2. Determine how many even-loving and how many odd-loving numbers there are.

(E. Novotná)
First, we determine the number of all even-loving numbers:

- If the digit 1 is in the hundreds place, the tens place can be any even digit $(0,2, 4,6,8)$ and the units place can be the same; this gives a total of $5 \cdot 5=25$ possibilities.
- If the digit 1 is in the tens place, the hundreds place can be any non-zero even digit $(2,4,6,8)$ and the units place can be any even digit; this gives us another $4 \cdot 5=20$ possibilities.
- If the digit 1 is in the units place, the hundreds place can be any non-zero even digit and the tens place can be any even digit; this gives us another $4 \cdot 5=20$ possibilities.

The total number of even-loving numbers is $25+20+20=65$.

Odd-loving numbers contain, in addition to 2, only odd digits. We can consider the same as when counting even-loving numbers, but in each case, we can count $5 \cdot 5=25$ possibilities (any odd digit $1,3,5,7,9$ can be in each free place). The total number of odd-loving numbers is $3 \cdot 25=75$.

Evaluation. 4 points for determining the number of even-loving numbers; 2 points for determining the number of odd-loving numbers.