How many five digit even natural numbers have exactly two 5s?

The last digit is 0, 2, 4, 6 or 8 -> 5 possibilities
Among the remaining 4 digits, there are two 5s.
If the other two digits are same, they can be selected in 9 ways and the four digits can be arranged in 4!/(2!*2!) = 6 ways
If the other two digits are different, they can be selected in 9*8 ways and the four digits can be arranged in 4!/2! = 12 ways
Total number of numbers = 5*(9*6+36*12) = 2430

Numbers starting with 0: If the other number among the four is also 0, the number of arrangements = 3
If the other number is not 0, it can be selected in 8 ways (0 and 5 should not be selected). The number of arrangements = 3
So, total number of numbers starting with 0 = 5*(3 + 8*3) = 27*5 = 135

Required number = 2430 - 135 = 2295