Final answer:
There are 130 positive integers less than 1000 divisible by 7 but not by 11, 648 positive integers with distinct digits, and 360 even positive integers with distinct digits.
Step-by-step explanation:
Positive Integers Divisible by 7 but Not by 11
To find how many positive integers less than 1000 are divisible by 7 but not by 11, we can calculate the number divisible by 7 and then subtract the number divisible by both 7 and 11. As 1000 divided by 7 gives approximately 142, there are 142 numbers less than 1000 that are multiples of 7.
Since 77 is the first common multiple of 7 and 11, and 1000 divided by 77 gives approximately 12, there are 12 numbers that are multiples of both. Hence, 130 positive integers are divisible by 7 but not by 11.
Positive Integers with Distinct Digits
There are 9 options for the first digit (1-9), 10 options for the second digit (0-9, minus the first digit), and 10 options for the third digit (0-9, minus the first two digits). So the total number is 9 * 9 * 8 = 648 integers with distinct digits.
Even Positive Integers with Distinct Digits
Even numbers end with 0, 2, 4, 6, or 8. For a three-digit number, there are 5 options for the last digit (0,2,4,6,8), 9 options for the first digit (1-9, excluding the one used in the last digit), and 8 options for the second digit (0-9, excluding the first and last digits).
So the total is 5 * 9 * 8 = 360 even integers with distinct digits.