$R _{3} \rightarrow R _{3}+ R _{1}$

$| A |=|\alpha+\beta+\gamma|\left|\begin{array}{ccc}\alpha & \beta & \gamma \\ \alpha^{2} & \beta^{2} & \gamma^{2} \\ 1 & 1 & 1\end{array}\right|$

$\Rightarrow| A |=(\alpha+\beta+\gamma)(\alpha-\beta)(\beta-\gamma)(\gamma-\alpha)$

$\because|\operatorname{adj} A |=| A |^{ n -1}$

$|\operatorname{adj}(\operatorname{adj} A )|=| A |^{( n -1)^{2}}$

$|\operatorname{adj}(\operatorname{adj}(\operatorname{adj}(\operatorname{adj} A )))|=| A |^{( n -1)^{4}}=| A |^{2^{4}}=| A |^{16}$

$\therefore(\alpha+\beta+\gamma)^{16}=2^{32} \cdot 3^{16}$

$(\alpha+\beta+\gamma)^{16}=\left(2^{2} \cdot 3\right)^{16}=(12)^{16}$

$\alpha+\beta+\gamma=12$

$\because \alpha, \beta, \gamma \in N$

$(\alpha-1)+(\beta-1)+(\gamma-1)=9$

number all tuples $(\alpha, \beta, \gamma)={ }^{11} C _{2}=55$

$1$ case for $\alpha=\beta=\gamma$

$12$ case when any two of these are equal

So, No. of distinct tuples $(\alpha, \beta, \gamma)$

$=55-13=42$