There are $15$ chairs arranged in a row. The position of $P$ is exactly at the middle, and $Q$ is at the $12\text{th}$ position when counting from the right end. How many chairs are located between $P$ and $Q$?

Let's determine the specific positions of $P$ and $Q$ in a standard left-to-right numbering system ($1$ to $15$):
1. **Position of $P$:** Since $P$ is in the exact middle of $15$ chairs, its position is:

$$\text{Position from Left}=\frac{15+1}{2}=8$$

2. **Position of $Q$:** $Q$ is $12\text{th}$ from the right end. The conversion formula to find the position from the left end is:

$$\text{Position from Left}=(\text{Total chairs}-\text{Position from Right})+1$$

$$\text{Position from Left for }Q=(15-12)+1=3+1=4$$

3. **Chairs between $P$ and $Q$:** $P$ is at position $8$ and $Q$ is at position $4$. The chairs between them occupy positions $5$, $6$, and $7$.

$$\text{Number of chairs}=8-4-1=3$$