We include all 3-digit even numbers that don't contain a 5 at all.

Case 1: the number does not contain a 5 at all.
We choose the first digit 8 ways [from {1,2,3,4,6,7,8,9}]
We choose the second digit 9 ways [from {0,1,2,3,4,6,7,8,9}]
We choose the third digit 5 ways [from {0,2,4,6,8}].

That's 8∙9∙5 = 360 ways

Case 2: the first digit is a 5 and the second digit is a 7.

They are 570, 572, 574, 576, and 578.

Total 360+5 = 365. 

[Note: the 2nd digit cannot be 5 because the third digit must be even, and
thus cannot be 7]

Edwin