How many integer numbers with four distinct digits exist that they are either additive or reductive (check my answer)

Question

This is a homework but I’m not sure I’m doing it right .

The question is : A) How many integer numbers with four distinct digits exist that they are either in increasing order (like 1234,3689,3679) or in decreasing order (like 8764,7410) ?

For additive part , This is what I did: I think for every four chosen digit between 1 to 9 , there are 4! Ways to arrange them as a four digit number , and only one of them has the property of being additive And if we want to consider 0 , it can’t be anywhere in our choosen 4 digits So we have $ 9\choose 4$ For reductive part 0 is involved, we can only place it as the last digit in our four digit number . So we have :${9\choose 4}+{9\choose 3}$

So the final answer for the question would be : $2{ 9\choose 4}+{9\choose 3}$

Please check my answers. And if I’m doing somthing wrong correct me.

Answer 1

Yes, it is correct. Your proof is also correct as it is. :)

Note: It was clarified in the comments that "additive" means "in increasing order" and "reductive" means in decreasing order, and that the digits can not be repeated.