The most restrictive digits are the first and last.  So we will choose
the last digit first and the first digit second.

The last digit can only be 0,2,4,6, or 8, in order for the number to be evem.
The first digit can only be 1,2,3,4,5,6,7,8, or 9 in order for the number to
be a five digit number.

IOW, the last digit must be even and the first digit cannot be 0.

If we choose 0 for the last digit, then we can choose the first
digit any of these 1,2,3,4,5,6,7,8, or 9.  That's 9 ways.

But if we don't choose 0 for the last digit, then we can only choose
the first digit 8 ways.  So we must break this problem into two
parts:

1.  When we choose 0 for the last digit.
2.  When we don't choose 0 for the last digit.

1.  Choose 0 for the last digit.  That's 1 way.
    Choose the first digit 9 ways.
    Choose the 2nd digit 8 ways.
    Choose the 3rd digit 7 ways.
    Choose the 4th digit 6 ways.

That's 1*9*8*7*6 = 3024 ways

2.  Choose 2,4,6, or 8 for the last digit. That's 4 ways.
    Choose the first digit 8 ways.
    Choose the 2nd digit 8 ways.
    Choose the 3rd digit 7 ways.
    Choose the 4th digit 6 ways.

That's 4*8*8*7*6 = 10752 ways

Total number of ways = 3024 + 10752 = 13776 ways.

Edwin