I can't say I like this problem much.

Firstly it is a little unclear; I assume that you are not allowed to reuse numerals (so 202020 is not a valid answer) but that should be explicit.

Secondly, there are 5*5*4*3*2 = 600 numbers. There is too high a ratio of writing to thinking here.

To do it,

*work through the legal first digits (that is, not 0) in increasing order. Within each of these:
*work through the legal second digits (may be 0, may not equal
first digit) in increasing order. Within each of these:
*work through the legal third digits (may be 0, may not equal
first or second digit) in increasing order. [Etc]

(2)
(20)
(203)
(2035)
203579
203597
(2037)
203759
203795
...
(205)
...
(207)
...
(209)
...
(23)
(230)
...
...
(3)
...

Good Hunting!
RD