Forgetful distribution

Algebra Level 4

When first introduced to multiplication in algebra, a common mistake made by students is to claim that \[ a \times ( b + c) = a\times b + c, \] because they forget to distribute \(a\) properly. For how many ordered triples of integers \( (a, b, c) \), each of which are between 0 and 10 inclusive, do we actually have \[ a \times ( b + c) = a\times b + c?\]