Power Tower Luck

Algebra Level 4

Tommy incorrectly thinks that \[\Large \color{blue}{a}^{\color{green}{b}^{\color{red}{c}}}= \left(\color{blue}{a}^{\color{green}{b}}\right)^{\color{red}{c}},\] while Jenny correctly knows that \[\Large \color{blue}{a}^{\color{green}{b}^{\color{red}{c}}}= \color{blue}{a}^{\left(\color{green}{b}^{\color{red}{c}}\right)}.\]

For how many of the 64 ordered triples of positive integers less than or equal to 4, \((\color{blue}{a},\color{green}{b},\color{red}{c}) \in \{1,2,3,4\}^3,\) will Tommy's incorrect method get the correct answer?

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