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 ____________.

The number of functions $$f$$, from the set $$\mathrm{A}=\left\{x \in \mathbf{N}: x^{2}-10 x+9 \leq 0\right\}$$ to the set $$\mathrm{B}=\left\{\mathrm{n}^{2}: \mathrm{n} \in \mathbf{N}\right\}$$ such that $$f(x) \leq(x-3)^{2}+1$$, for every $$x \in \mathrm{A}$$, is ___________.

Let for the $$9^{\text {th }}$$ term in the binomial expansion of $$(3+6 x)^{\mathrm{n}}$$, in the increasing powers of $$6 x$$, to be the greatest for $$x=\frac{3}{2}$$, the least value of $$\mathrm{n}$$ is $$\mathrm{n}_{0}$$. If $$\mathrm{k}$$ is the ratio of the coefficient of $$x^{6}$$ to the coefficient of $$x^{3}$$, then $$\mathrm{k}+\mathrm{n}_{0}$$ is equal to :

$$ \frac{2^{3}-1^{3}}{1 \times 7}+\frac{4^{3}-3^{3}+2^{3}-1^{3}}{2 \times 11}+\frac{6^{3}-5^{3}+4^{3}-3^{3}+2^{3}-1^{3}}{3 \times 15}+\cdots+ \frac{30^{3}-29^{3}+28^{3}-27^{3}+\ldots+2^{3}-1^{3}}{15 \times 63}$$ is equal to _____________.