CEMC Pascal, 2007

$P, Q, R, S$, and $T$ are five different integers between $2$ and $19$ inclusive.

$\bullet$ $P$ is a two-digit prime number whose digits add up to a prime number.

$\bullet$ $Q$ is a multiple of $5$.

$\bullet$ $R$ is an odd number, but not a prime number.

$\bullet$ $S$ is the square of a prime number.

$\bullet$ $T$ is a prime number that is also the mean (average) of $P$ and $Q$.

Which number is the largest?

(A) $P$ (B) $Q$ (C) $R$ (D) $S$ (E) $T$