Counting

Since we are asked for the number of positive 4-digit integers with at least 5 or 8 in it, we can find this by finding the total number of 4-digit integers and subtracting off those which do not have any 5s or 8s as digits.

The total number of 4-digit integers is 9×10×10×10 = 9000, since we have 10 choices for each digit except the first (which can’t be 0).

Similarly, the total number of 4-digit integers without any 5 or 8 is 7 × 8 × 8 × 8 = 3584 

Therefore, the total number of positive 4-digit integers that have at least one 5 or 8 in their decimal representation is 9000 – 3584 = 5416.