Emoticon

Lets divide the possibilities in two parts:-
Other part which is a collection of numbers which do not end with 0.
One part which is a collection of numbers which end with 0.

Part 1:-
If the number does not end with 0, the units digit can be filled in with the other even numbers which is in $4$ ways. The ten thousandth digit can be filled in $8$ ways becayse we cant place 0 and one digit has already been used up. Then the thousandth digit can be filled in $8$ ways because 2 numbers has been used up and 0 can be used. The hundredth digit can be filled in $7$ ways because 3 digits have already been used up. Lastly, the tenth digit can be filled in $6$ ways because 4 digits have been used up. So, the number of even 5 digit numbers without repitition of digits which do not end with 0 that can be formed = $8×8×7×6×4 = 10752$ .

Part 2:-
If the number ends with $0$ then we can fill the units digit in only $1$ way. Then we can fill the ten thousandth digit in $9$ ways. Then the thousandth in $8$ ways, the hundredth in $7$ ways and the tenth in $6$ ways. So, the number of 5-digit even numbers without repitition of digits which end with $0$ can be formed in $9×8×7×6×1 = 3024$ .

Since we can make numbers which either fall in part 1 $\text{or}$ part 2, we add the results. So, the final answer is $10752 + 3024 = \color{#69047E}{\boxed{13776}}$ .

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Alternatively:-
Total number of 5 - digit odd numbers without repitition of digits can end in $5$ ways. The ten thousandth digit can be filled in $8$ ways because 0 cant be used and 1 number is already used up. So, the thousandth digit can be filled in $8$ ways because 2 numbers are used up and 0 can be used now. The hundredth digit can be filled in $7$ ways and the tenth digit can be filled in $6$ ways which give the result as $8×8×7×6×5 = 13440$ .

Total 5-digit numbers without repitition of digits that can have their ten thousandth digit in $9$ ways because 0 cant be used. Then the thousandth digit can be filled in $9$ ways again because 1 number is used up and 0 can be used. Similarly, the hundredth digit can be filled in $8$ ways and the tenth digit can be filled in $7$ ways and the units digit in $6$ ways which gives the result as $9×9×8×7×6 = 27216$ .

Now 5-digit numbers whose digits do not repeat are either even or odd. So, the 5-digit even numbers whose digits do not repeat can be obtained by subtracting the number of 5-digit odd numbers whose digits do not repeat from the total number of 5-digit numbera whose digits do not repeat. So, the answer is $27216 - 13440 = \color{#69047E}{\boxed{13776}}$ .