Understanding the Problem
We have a group of 65 people where:
- Every person likes at least one of two activities: sky-diving or car-driving.
- 48 persons like sky-diving.
- 27 persons like car-driving.

We need to find out how many persons like only sky-diving.
Visualizing the Problem
This is a classic problem that can be visualized using a Venn diagram, where we have two circles representing:
1. People who like sky-diving.
2. People who like car-driving.

The overlapping area represents people who like both activities.
Defining the Variables
Let's define:
- Let $S$ be the set of people who like sky-diving.
- Let $C$ be the set of people who like car-driving.
- $|S|=48$ (number of people who like sky-diving).
- $|C|=27$ (number of people who like car-driving).
- Total number of people, $\(|S \cup C|=65\)$.

We need to find the number of people who like only sky-diving, which is $\(|S|-|S \cap C|\)$, where $\(|S \cap C|\)$ is the number of people who like both.

$$
\(|S \cup C|=|S|+|C|-|S \cap C|\)
$$


Plugging in the known values:

$$
\(\begin{gathered}
65=48+27-|S \cap C| \\
65=75-|S \cap C| \\
|S \cap C|=75-65=10
\end{gathered}\)
$$


So, 10 people like both sky-diving and car-driving.
Finding Only Sky-diving
Now, the number of people who like only sky-diving is:

$$
\(\text { Only sky-diving }=|S|-|S \cap C|=48-10=38\)
$$


Verifying the Answer
Let's check if this makes sense:
- Only sky-diving: 38
- Only car-driving: $\(|C|-|S \cap C|=27-10=17\)$
- Both: 10

Total: $38+17+10=65$, which matches the given total.
Checking the Options
The number of persons who like only sky-diving is 38 . Looking at the options:
- A. 17
- B. 27
- C. 38
- D. 40
- E. 45

The correct answer is $\(\mathbf{C . ~} \mathbf{3 8}\)$.