Consider a matrix $$A=\left[\begin{array}{ccc}\alpha & \beta & \gamma \\ \alpha^{2} & \beta^{2} & \gamma^{2} \\ \beta+\gamma & \gamma+\alpha & \alpha+\beta\end{array}\right]$$, where $\alpha, \beta, \gamma$ are three distinct natural numbers.

If $$\frac{\operatorname{det}(\operatorname{adj}(\operatorname{adj}(\operatorname{adj}(\operatorname{adj} A))))}{(\alpha-\beta)^{16}(\beta-\gamma)^{16}(\gamma-\alpha)^{16}}=2^{32} \times 3^{16}$$, then the number of such 3 - tuples $(\alpha, \beta, \gamma)$ is ____________.

<p>$$\det (A) = \left| {\matrix{ \alpha & \beta & \gamma \cr {{\alpha ^2}} & {{\beta ^2}} & {{\gamma ^2}} \cr {\beta + \gamma } & {\gamma + \alpha } & {\alpha + \beta } \cr } } \right|$$</p> <p>${R_3} \to {R_3} + {R_1}$</p> <p>$$ \Rightarrow (\alpha + \beta + \gamma )\left| {\matrix{ \alpha & \beta & \gamma \cr {{\alpha ^2}} & {{\beta ^2}} & {{\gamma ^2}} \cr 1 & 1 & 1 \cr } } \right|$$</p> <p>$\therefore$ $$\det (A) = (\alpha + \beta + \gamma )(\alpha - \beta )(\beta - \gamma )(\gamma - \alpha )$$</p> <p>Also, $\det (adj\,(adj\,(adj\,(adj\,(A)))))$</p> <p>$= {(\det (A))^{{2^4}}} = (\det {(A)^{16}}$</p> <p>$\therefore$ $${{{{(\alpha + \beta + \gamma )}^{16}}{{(\alpha - \beta )}^{16}}{{(\beta - \gamma )}^{16}}{{(\gamma - \alpha )}^{16}}} \over {{{(\alpha - \beta )}^{16}}{{(\beta - \gamma )}^{16}}{{(\gamma - \alpha )}^{16}}}} = {(4.13)^{16}}$$</p> <p>$\Rightarrow \alpha + \beta + \gamma = 12$</p> <p>$\Rightarrow (\alpha ,\beta ,\gamma )$ distinct natural triplets</p> <p>$= {}^{11}{C_2} - 1 - {}^3{C_2}(4) = 55 - 1 - 12$</p> <p>$= 42$</p>