P: Set of Rational numbers are countable. Rational numbers are of the form $\frac{p}{q}$ where $p, q$ are integers. Enumeration procedure, take $p+q$ and write down all possible values(positive and negative).
Q: Set of functions from $\left \{ {0,1} \right \}$ to $N$. There are $N^2$ such functions. Hence countable.
R: Set of functions from $N$ to $\left \{ {0,1} \right \}$. There are $2^N$ such functions. Important theorem $\Rightarrow$
If a set $S$ is countable, then $\mathbb{P}(S)$ i.e $2^S$ is uncountable.
Hence, statement R is uncountable.
S: Set of finite subsets of $N$. They are countable. Important theorem $\Rightarrow$
Every subset of a countable set is either countable or finite.
Hence, Option (D).