Problem #WSP-000267

Problem

Paloma wrote digits from $0$ to $9$ in each of the $9$ dots below, using each digit at most once. Since there are $9$ dots and $10$ digits, she must have missed one digit.

In the triangles, Paloma started writing either the three digits at the corners added together (the sum), or the three digits at the corners multiplied together (the product). She gave up before finishing the final two triangles.

What numbers could Paloma have written in the interior of the red triangle? Demonstrate that you’ve found all of the possibilities.