According to the answer sheet:

Case 1:
If number 2 comes first in the 5-digit number, we can choose remaining numbers in 8\*7\*6 different ways.

Case 2: 
The number of ways to place number 2 and 3 after each other so that number 2 doesn't appear first can be formed in 3\*7\*7\*6 ways, because the numbers 2 and 3 can be placed on 3 different places. And we can choose the first number in 7 different ways (can't be 0), and the remaining numbers in 7\*6 different ways.

So, I understand the first case. What I don't get is the second case.
Let's say I put the sequence of numbers like this: "\_ \_ \_ 2 3" and decide to pick the third number first (8 ways), then the second number (7 ways) and lastly the first number (5 ways). I won't get the same result, so my question is: why do I have to do this in order, starting from the first digit?