THe first digit can be $5,6,7,8,9$. Those are four posibilities.
The second digit can be any of the ten $0,1,2..., 9$ (for some inexplicable reason you didn't include $0$) but the second digit must be different from the first. So there are $9$ options.
The third digit must be different from the first two so there are $8$ options.
The fourth must be different for the first three so so there are $7$ options.
The last digit must be even so it is $0,2,4,6,8$ and it must be different that the first four and ... we have no idea how many of the first four are even or not. So we are found in the Alps. Dang.
Do two cases. Either the first number is even $6,8$ (2 options) or it is $5,7,9$ (3 options).
That last digit must be even so if the first is even then the second must be different so there are $4$ options because it must be different. If the first is not even there are $5$ options.
The second digit is different that the first or last so there are $8$ options.
The third digit must be different than the other three so there are $7$ options.
So if the first digit is even there are $2*4*8*7$. And if the first digit is not even there are $3*5*8*7$.
So there are $2*4*8*7 + 3*5*8*7 = (2*4 + 3*5)*8*7 = (8+15)*56 = 23*56= 1288$ such numbers.