The domain of the definition of the function $$y(x)$$ is given by the equation $$2^x+2^y=2$$ is

If $$\mathbf{a}=\frac{1}{\sqrt{10}}(3 \hat{\mathbf{i}}+\hat{\mathbf{k}}), \mathbf{b}=\frac{1}{7}(2 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}-6 \hat{\mathbf{k}})$$, then the value of $$(2 \mathbf{a}-\mathbf{b}) \cdot[(\mathbf{a} \times \mathbf{b}) \times(\mathbf{a}+2 \mathbf{b})]$$ is

$$y=\frac{\sqrt[3]{1+3 x} \sqrt[4]{1+4 x} \sqrt[5]{1+5 x}}{\sqrt[7]{1+7 x} \sqrt[8]{1+8 x}} \text {. Then, } \frac{d y}{d x} \text { at } x=0$$ is

Five students are selected from $$n$$ students such that the ratio of number of ways in which 2 particular students are selected to the number of ways 2 particular students not selected is $$2: 3$$. Then, the value of $$n$$ is