Example 1-2 Find the number of positive integers less than 10000 that contain the digit 1.
Positive integers less than 10000 can be seen as formed by taking four numbers from $0,1,2, \cdots, 8,9$. But 0000 is not within the range of positive integers, so the number of positive integers less than 10000, according to the multiplication rule, should be $10^{4}-1=9999$.
The number of four-digit numbers that do not contain 1 is
$$
9^{4}-1=6561-1=6560
$$

Therefore, the number of numbers less than 10000 that contain 1 is
$$
9999-6560=3439
$$

This solution does not directly find the number of numbers containing 1, but rather finds the number of numbers that do not contain 1. Subtracting the numbers that do not contain 1 from the total leaves the numbers that contain 1.