A survey was conducted using 200 university students.

   Out of 140 who like Peradeniya also, 45 like Peradeniya only.

 There were 7 people who disliked all the three universities.

Of the 29 who preferred both Colombo and Kelaniya universities, 5 did not like Peradeniya.

Out of the 88 who did not like Colombo, 16 preferred Peradeniya and Kelaniya

1. Copy the given ten figure in the answer sheet and mark the above information there.

ii. How many students have applied to Colombo University only?

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

A survey was conducted using 200 university students. Out of 140 who like Peradeniya also, 45 like Peradeniya only. There were 7 people who disliked all the three universities. Of the 29 who preferred both Colombo and Kelaniya universities, 5 did not like Peradeniya. Out of the 88 who did not like Colombo, 16 preferred Peradeniya and Kelaniya. How many students have applied to Colombo University only?

Solve a set theory problem using a Venn diagram with 200 university students. Learn how to calculate the number of students who applied to Colombo University only, using key concepts like union, intersection, and the inclusion-exclusion principle. Suitable for Grades 9-12.

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