Number of 3-digit integers less than 180 with exactly different digits in their representation: 176

To solve this problem, we need to find the number of 3-digit integers less than 180 that have exactly different digits in their representation. Let's break down the problem step by step:

Step 1: Understanding the problem
We are given the condition that the 3-digit integers must have exactly different digits in their representation. This means that all three digits in the number should be different from each other.

Step 2: Analyzing the range
We need to find the 3-digit integers less than 180. The given range is from 100 to 179, inclusive.

Step 3: Analyzing the hundreds digit
The hundreds digit can take values from 1 to 9 since the number needs to be a 3-digit integer. Therefore, there are 9 possible choices for the hundreds digit.

Step 4: Analyzing the tens digit
The tens digit can take values from 0 to 9, excluding the value of the hundreds digit. If the hundreds digit is 1, the tens digit can be any value from 0 to 9 except 1. If the hundreds digit is 2, the tens digit can be any value from 0 to 9 except 2, and so on. Similarly, for each hundreds digit, there are 9 possible choices for the tens digit.

Step 5: Analyzing the units digit
The units digit can take values from 0 to 9, excluding the values of the hundreds and tens digits. For each combination of hundreds and tens digits, there is only 1 possible choice for the units digit.

Step 6: Calculating the total number of 3-digit integers
To calculate the total number of 3-digit integers meeting the given condition, we multiply the number of choices for each digit. So, the total number of 3-digit integers is:
9 (choices for the hundreds digit) * 9 (choices for the tens digit) * 1 (choice for the units digit) = 81

Step 7: Adjusting for the given range
We need to consider that the integers should be less than 180. Out of the 81 possible combinations, we need to exclude the ones that are greater than or equal to 180. We can see that there are 4 combinations where the hundreds digit is greater than or equal to 2 (200, 210, 220, and 230). For each of these combinations, there are 9 choices for the tens digit and 1 choice for the units digit. So, the number of 3-digit integers greater than or equal to 200 is 4 * 9 * 1 = 36.

Therefore, the number of 3-digit integers less than 180 with exactly different digits in their representation is 81 - 36 = 45.

Step 8: Final answer
The final answer is 45.