Let
$$\mathrm{p}$$ : Ramesh listens to music.
$$\mathrm{q}$$ : Ramesh is out of his village.
$$\mathrm{r}$$ : It is Sunday.
$$\mathrm{s}$$ : It is Saturday.
Then the statement "Ramesh listens to music only if he is in his village and it is Sunday or Saturday" can be expressed as
Let the coefficients of the middle terms in the expansion of $$\left(\frac{1}{\sqrt{6}}+\beta x\right)^{4},(1-3 \beta x)^{2}$$ and $$\left(1-\frac{\beta}{2} x\right)^{6}, \beta>0$$, respectively form the first three terms of an A.P. If d is the common difference of this A.P. , then $$50-\frac{2 d}{\beta^{2}}$$ is equal to __________.
A class contains b boys and g girls. If the number of ways of selecting 3 boys and 2 girls from the class is 168 , then $$\mathrm{b}+3 \mathrm{~g}$$ is equal to ____________.
Let the tangents at the points $$\mathrm{P}$$ and $$\mathrm{Q}$$ on the ellipse $$\frac{x^{2}}{2}+\frac{y^{2}}{4}=1$$ meet at the point $$R(\sqrt{2}, 2 \sqrt{2}-2)$$. If $$\mathrm{S}$$ is the focus of the ellipse on its negative major axis, then $$\mathrm{SP}^{2}+\mathrm{SQ}^{2}$$ is equal to ___________.