Tricky question!

Here's why: If we first choose zero for the units digit, then the hundreds digit can be 1,2,3,4,5,6,7,8 or 9 (9 possible values)

But, if we DON'T first choose zero for the units digit, then the hundreds digit cannot be zero, since the resulting number wouldn't be a 3-digit number (e.g., 046) is not a 3-digit number). Also, the hundreds digit cannot be the same as the units digit (since all 3 digits are DIFFERENT). So, if we DON'T first choose zero for the units digit, then the hundreds digit can have only 8 possible values.

Given this, we must treat each case separately.

Case A: We choose zero for the units digit
Take the task of creating 3-digit integers, and break it into stages.

We’ll begin with the most restrictive stage.

Stage 1: Select a units digit
Since the units digit MUST be 0 for this case, we can complete stage 1 in 1 way

Stage 2: Select a hundreds digit
The hundreds digit can be 1,2,3,4,5,6,7,8 or 9
So, we can complete stage 2 in 9 ways

Stage 3: Select a tens digit
The tens digit can be any of the 10 digits EXCEPT for 0 and the number selected for the hundreds digit
So, we can complete stage 3 in 8 ways

By the Fundamental Counting Principle (FCP), we can complete all 3 stages (and thus create a 3-digit number) in (1)(9)(8) ways (= 72 ways)
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Case B: We DO NOT choose zero for the units digit

Stage 1: Select a units digit
Since the units digit must be EVEN (but can't be 0), the units digit can be (2,4,6 or 8)
So, we can complete stage 1 in 4 ways

Stage 2: Select a hundreds digit
There are 10 digits in total.
HOWEVER, the hundreds digit cannot be zero, AND the hundreds digit cannot be the same as the units digit (since all 3 digits are DIFFERENT)
So, we can complete stage 2 in 8 ways

Stage 3: Select a tens digit
The tens digit can be any of the 10 digits EXCEPT for the two digits selected for the hundreds and units place
So, we can complete stage 3 in 8 ways

By the Fundamental Counting Principle (FCP), we can complete all 3 stages (and thus create a 3-digit number) in (4)(8)(8) ways (= 256 ways)
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So, the TOTAL number of possible 3-digit numbers = 72 + 256 = 328

Answer: 328

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