Well, none of the digits can be 0 because the product is than going to be 0. You can't have more than 2 1's either.

To get started, we turn to the factorization of 10,000. It factors out as 2^4 * 5^4. This means that the only possible combinations of numbers are:

1,2,8,5,5,5,5 -> 105 ways to arrange this

2,2,4,5,5,5,5 -> 105 ways to arrange this

1,4,4,5,5,5,5 -> 105 ways to arrange this

1,1,8,5,5,5,5 -> 105 wasy to arrange this

So the number of different possible codes is 105+105+105+105 = **420**** possible codes**.