How many three-digit numbers in base 10 are three-digit numbers in base 8 but not in base 7?

To solve this problem, we need to understand the number systems in different bases.

1. Base 10: This is the decimal system, which we use in everyday life. A three-digit number in base 10 ranges from 100 to 999.

2. Base 8: This is the octal system. A three-digit number in base 8 ranges from 100 to 777 (in base 10, this is 64 to 511).

3. Base 7: This is the septenary system. A three-digit number in base 7 ranges from 100 to 666 (in base 10, this is 49 to 342).

We are looking for numbers that are three digits in both base 10 and base 8, but not in base 7. This means we are looking for numbers that fall within the range of 100 to 511 in base 10, but are greater than 342.

So, the numbers we are looking for are from 343 to 511.

To find the total number of these numbers, we subtract the smaller number from the larger number and then add 1 (because both 343 and 511 are included).

So, the total number of three-digit numbers in base 10 that are three-digit numbers in base 8 but not in base 7 is 511 - 343 + 1 = 169.