As you say, a number is even if and only its "ones" place is even as you say.
I would count the number of one-digit, two digit, three digit, four digit, and five digit numbers.
(Do you see why I do not include six and seven digit numbers?)
There are three such numbers with one digit: 2, 4, and 6. (0 is not a natural number.)
For two digit numbers, once we have one of 2, 4, and 6 as the ones digit, there are five possible 10's digits (again, 0 cannot be the 10's digit in a two digit number.) That gives 3(5)= 15 such numbers. If 0 is the one's digit, then there are 6 possible digits for the 10's digit so that is another 6 giving 6+ 15= 21 two digit even numbers,
For three digit numbers, we can have anyone of those 15 two digit numbers that do NOT have a 0 with the remaining four non-zero digits so 4(15)= 60. Of the six that do have a 0, we can take any of the remaining 5 digits as the new digit, The are 5(6)= 30 such numbers so 60+ 30= 90 such three digit numbers.
Similarly, of the 60 three digit numbers that do NOT have a 0 we can add any of the remaining three non-zero digits so 60(3)= 180. Of the 30 numbers that do have a 0 we can add any of the remaining four non-zero digits. There are 30(4)= 120 such numbers so 180+ 120= 300 such four digit numbers.
Finally, of the 180 four digit numbers that do NOT have a 0 we can add any or the remaining 2 non-zero digits so (180)(2)= 360. Of the 120 four digit numbers we also add only the two non-zero digits because the first digit in a number cannot be 0, That is (120)(2)= 240 so there are 360+ 240= 600 such five digit numbers.
That gives a total of 3+ 21+ 90+ 300+ 600= 1014 even numbers, under 100000, using those digits at most once.
