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

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

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