The number 2014 has 4 distinct digits; 1 digit is odd, and 3 digits are even, of which one is zero.

How many 4-digit numbers (the first cannot be 0) have these properties?

The answer is 540.

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The mistake that I made for my first try was that for some reason I neglected the zero condition.

Anyway straight forward counting:

first we place the zero, $3$ choices(not the first digit) then the odd number, $3$ choices for position and $5$ choices for number. finally the even number, $4*3$ ways to fill the vacant spots.

Together $3*3*5*4*3=\boxed {540}$ is our answer.