1. Interpret the condition

There are $3$ sections, each containing $5$ questions.

The candidate must choose a total of $5$ questions such that at least one question is chosen from each section.

So if the numbers of questions chosen from the three sections are $x,y,z$, then:

$x+y+z=5, \quad x,y,z\ge 1$

2. Find all possible distributions

We need positive integer solutions of:

$x+y+z=5$

The possible distributions are:

• $(3,1,1)$ and its permutations
• $(2,2,1)$ and its permutations

So we count each case separately.

3. Case 1: Distribution $(3,1,1)$

Choose which section contributes $3$ questions:

$3 \text{ ways}$

From that section, choose $3$ questions out of $5$:

$\binom{5}{3}=10$

From each of the other two sections, choose $1$ question out of $5$:

$\binom{5}{1}=5 \quad \text{each}$

Thus total ways for this case:

$3 \cdot \binom{5}{3}\cdot \binom{5}{1}\cdot \binom{5}{1}$

$=3\cdot 10\cdot 5\cdot 5=750$

4. Case 2: Distribution $(2,2,1)$

Choose which section contributes $1$ question:

$3 \text{ ways}$

From the two sections contributing $2$ questions each, choose $2$ out of $5$:

$\binom{5}{2}=10$

From the remaining section, choose $1$ question:

$\binom{5}{1}=5$

Thus total ways for this case:

$3 \cdot \binom{5}{2}\cdot \binom{5}{2}\cdot \binom{5}{1}$

$=3\cdot 10\cdot 10\cdot 5=1500$

5. Add both cases

$750+1500=2250$

So the required number of ways is:

$\boxed{2250}$

6. Check with options

Option $A$ is $2250$, so the correct option is:

$\boxed{A}$

7. Comparison with stored answer

Stored correct answer = $A$

Our derived answer = $A$

Hence, they agree.