$\text{A}$, $\text{B}$, $\text{C}$, $\text{D}$ and $\text{E}$ are sitting in a row. $\text{B}$ is between $\text{A}$ and $\text{E}$. To find out who among them is in the middle, which of the following statements $\text{P}$ and $\text{Q}$ is/are sufficient?
$\text{P.}$ $\text{A}$ is left of $\text{B}$ and right of $\text{D}$.
$\text{Q.}$ $\text{C}$ is at the right end.

We are given that 5 persons ($\text{A, B, C, D, E}$) are sitting in a row, and $\text{B}$ sits somewhere between $\text{A}$ and $\text{E}$.
Let's evaluate Statement $\text{P}$ alone:
$\text{A}$ is to the left of $\text{B}$ and to the right of $\text{D}$, which gives the order $\text{D}\dots \text{A}\dots \text{B}$. Since $\text{B}$ is between $\text{A}$ and $\text{E}$, $\text{E}$ must be to the right of $\text{B}$, resulting in $\text{D}\dots \text{A}\dots \text{B}\dots \text{E}$. This leaves the position of $\text{C}$ undetermined; $\text{C}$ could be at either end. Thus, statement $\text{P}$ alone is insufficient.

Let's evaluate Statement $\text{Q}$ alone:
$\text{C}$ is at the right end ($\_\_\_\_\text{C}$). Without information about the remaining letters, their exact arrangement cannot be determined. Thus, statement $\text{Q}$ alone is insufficient.

Combining Statements $\text{P}$ and $\text{Q}$:
From $\text{P}$, the relative sequence is $\text{D}-\text{A}-\text{B}-\text{E}$. From $\text{Q}$, $\text{C}$ occupies the fifth position on the extreme right. Merging these constraints yields a unique linear arrangement from left to right:

$$\text{D, A, B, E, C}$$

In this arrangement, $\text{B}$ is exactly in the middle (third position). Therefore, both statements $\text{P}$ and $\text{Q}$ together are necessary and sufficient.