Final answer:
To find the number of four-digit positive integers that have exactly 3 identical numbers, we can break down the problem into cases. There are 252 such numbers.
Step-by-step explanation:
To find the number of four-digit positive integers that have exactly 3 identical numbers, we can break down the problem into cases.
Case 1: The three identical numbers are the first three digits. In this case, the first digit can be chosen in 9 ways (excluding 0) and the fourth digit can be chosen in 9 ways (including 0 since it is not in the first three digits). So, there are 9 * 9 = 81 such numbers.
Case 2: The three identical numbers are the last three digits. This case is identical to Case 1, so there are also 81 numbers.
Case 3: The three identical numbers are the middle three digits. In this case, the first digit can be chosen in 10 ways (including 0 since it is not in the middle three digits) and the fourth digit can be chosen in 9 ways (excluding 0). So, there are 10 * 9 = 90 such numbers.
Adding the numbers from all three cases, the total number of four-digit positive integers with exactly 3 identical numbers is 81 + 81 + 90 = 252.