Common silly mistake

Algebra Level 5

A common mistake made by students is to claim that

$\sqrt{a^2+b^2} = \sqrt{a^2} + \sqrt{b^2} .$

How many ordered triples of integers $(a, b, c)$ are there, such that each of $a,b,c$ are integers from 0 to 9 inclusive, and

$\sqrt{ a^2 + b^2} - \sqrt{a^2 + c^2} = \sqrt{b^2} - \sqrt{c^2} ?$