Let's break down the information and use set theory to solve the problem, specifically using a Venn diagram. We are dealing with three universities: Peradeniya (P), Colombo (C), and Kelaniya (K). Let's denote the total number of students as $U = 200$.

We are given the following information:

1. There are 140 students who like Peradeniya. Of these, 45 like Peradeniya only.

   • So, the total number of students who like Peradeniya, $|P| = 140$, and the students who like Peradeniya only are 45.

2. There are 7 students who dislike all three universities.

   • This means 193 students (200 - 7) like at least one university.

3. Of the 29 students who like both Colombo and Kelaniya, 5 do not like Peradeniya.

   • This means that $|C \cap K| = 29$, but only 24 of them like Peradeniya, i.e., $|C \cap K \cap P| = 24$.

4. Out of the 88 students who do not like Colombo, 16 prefer both Peradeniya and Kelaniya.

   • This means that 16 students like Peradeniya and Kelaniya but not Colombo, i.e., $|P \cap K \setminus C| = 16$.

Venn Diagram Setup

We can now represent this in the Venn diagram with overlapping circles for $P$, $C$, and $K$.

Let's denote the number of students who like:

• $P \cap C \cap K$ as $x$, the students who like all three universities.
• $P \cap C \setminus K$ as $y$, the students who like Peradeniya and Colombo but not Kelaniya.
• $P \cap K \setminus C = 16$ (already given).
• $C \cap K \setminus P = 5$ (already given).
• $P \setminus (C \cup K) = 45$ (already given).

Now we can use the information above to find these numbers.

Step-by-Step Calculation:

1. Students who like all three universities ($x$):

   • Since $|C \cap K| = 29$ and 5 students who like both Colombo and Kelaniya do not like Peradeniya, the remaining 24 students must like all three universities. Thus, $x = 24$.

2. Students who like Peradeniya and Colombo but not Kelaniya ($y$):

   • The total number of students who like Peradeniya is 140. This includes:
     • 45 students who like only Peradeniya.
     • 24 students who like all three universities ($P \cap C \cap K$).
     • 16 students who like Peradeniya and Kelaniya but not Colombo.
   • Thus, $140 = 45 + x + y + 16$, and substituting $x = 24$, we get: $140 = 45 + 24 + y + 16$ Simplifying, $y = 55$.

3. Total number of students who like Colombo only:

   • The total number of students who like Colombo is $|C| =$ students in $C \cap P$, $C \cap K$, and $C \setminus (P \cup K)$.
   • We already know $C \cap K = 29$, so the total students who like Colombo only will be the students left after accounting for overlaps.

Now, to finalize how many applied to Colombo only, we