If the smallest digit were 0 for the product of the digits would be 0.
So 0 is eliminated.

The smallest digit cannot be 1 for the product of the other two digits 
would have to be 216, and that's impossible.  So 1 is eliminated

The smallest digit cannot be 2 as because 216/2 = 108
and so the other two would have to have product 108 and no two digits
have product that large.

If 3 is its smallest digit, then since 216/3 = 72, the other two digits
would have to have product 72, and that is only possible with digits
8 and 9, but 389, is not possible because 3+8+9=20, not 19.

If 4 is its smallest digit, then since 216/4 = 54, the other two digits
would have to have product 54, and that is only possible with digits
6 and 9, 4+6+9=19.   So 469 is an answer since the digits are ascending.

But we need to rule out any other answers.

5 cannot be the smallest digit because 216 is not a multiple of 5.

If 6 is its smallest digit, then since 216/6 = 36, the other two digits
would have to have product 36, and since we've ruled out 4, the only 
possible way is with the other two digits being 6 and 6.  but 666 is not
possible for two reasons, the digits are not ascending and because 6+6+6=18,
not 19.

The only answer is 469.   

Edwin