In distinct odd-town, the inhabitants want to number their houses with 3-digit positive integers that are odd, which have all distinct digits. What is the maximum number of houses in odd-town?
320
To find the maximum number of houses in odd-town, we need to consider the possible combinations of 3-digit positive odd integers with distinct digits.
The first digit in a 3-digit odd number can be chosen from the set {1, 3, 5, 7, 9}. Since we want distinct digits, there are 5 choices for the first digit.
The second digit can be chosen from the set {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}, excluding the digit already chosen for the first digit. Therefore, there are 9 choices for the second digit.
Finally, the third digit can be chosen from the set {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}, excluding the two digits already chosen for the first and second digits. Thus, there are 8 choices for the third digit.
Using the multiplication principle, the total number of 3-digit positive odd integers with distinct digits is calculated as follows:
Number of choices for the first digit × Number of choices for the second digit × Number of choices for the third digit = 5 × 9 × 8 = 360
Therefore, the maximum number of houses in odd-town is 360.