Let N be the set of natural numbers. Consider the following sets.

$$\,\,\,\,\,\,\,\,$$ $$P:$$ Set of Rational numbers (positive and negative)
$$\,\,\,\,\,\,\,\,$$ $$Q:$$ Set of functions from $$\left\{ {0,1} \right\}$$ to $$N$$
$$\,\,\,\,\,\,\,\,$$ $$R:$$ Set of functions from $$N$$ to $$\left\{ {0,1} \right\}$$
$$\,\,\,\,\,\,\,\,$$ $$S:$$ Set of finite subsets of $$N.$$

Which of the sets above are countable?

Which one of the following is a closed form expression for the generating function of the sequence $$\left\{ {{a_n}} \right\},$$ where $${a_n} = 2n + 3$$ for all $$n = 0,1,2,....?$$

The chromatic number of the following graph is _______.

Let $$G$$ be a finite group on $$84$$ elements. The size of a largest possible proper subgroup of $$G$$ is ________.