One attempt, just top of my head.
The total number of numbers greater than 10,000 using digits 0,2,4,6,8 is = $4*4*3*2 = 96$
Not to find the sum,
One's place:
0 will occur : 4*3*2*1 times = 24 times, sum =0
2 will occur: 3*3*2*1 times = 18 times, sum = 36
similarly 4 will occur 18 times sum = 72
6 also 18 times, sum = 108
and 8 also 18 times, sum = 144
total sum = 360, unit place has 0 and carry 36
-----------
tens place:
Same as above: total sum = 360+36 = 396.
one place = 6, carry 39
----
Hundered's place
Same, total sum = 360+39 = 399
Hundereds place = 9, carry 39
---
thousand same, total sum = 360+39 = 399
thousand place = 9, carry 39
---
ten thousand place
same, total sum = 360+39 = 399,
ten thousand place 9, carry 39
----- finally it will be different
now it will be different
2 will be: 24 times, total = 48
4 will be 24 times, total = 96
6 will be 24 times,total = 144
and 8 will be also 24 times = 192
total sum = 480, carry = 39 = 519
--------------------
and hence sum is: 519,9,9,9,6,0 = 5199960.
