A survey was conducted among 300 students. If was found that 125 students like play cricket, 145 students like to play football and 90 students like to play tennis, 32 students like to play exactly two games out of the three games.
How many students like to play all the three games?

Options

A. 14
B. 21
C. 28
D. 35

Explanation

To solve the problem, we need to determine how many students like to play all three games: cricket, football, and tennis. We will use the principle of inclusion-exclusion.

### Step-by-Step Solution:

1. **Define the Sets:**
- Let \( A \) be the set of students who like cricket.
- Let \( B \) be the set of students who like football.
- Let \( C \) be the set of students who like tennis.

From the problem, we have:
- \( |A| = 125 \)
- \( |B| = 145 \)
- \( |C| = 90 \)

2. **Total Students:**
- The total number of students surveyed is \( 300 \).

3. **Using the Inclusion-Exclusion Principle:**
- According to the principle, the total number of students can be expressed as:
\[ |A \cup B \cup C| = |A| + |B| + |C| - |A \cap B| - |B \cap C| - |A \cap C| + |A \cap B \cap C| \]
- We know that \( |A \cup B \cup C| = 300 \).

4. **Substituting Known Values:**
- We also know that \( 32 \) students like exactly two games. This means:
\[ |A \cap B| + |B \cap C| + |A \cap C| - 3|A \cap B \cap C| = 32 \]
- Let \( x = |A \cap B \cap C| \) (the number of students who like all three games).

5. **Setting Up the Equations:**
- From the inclusion-exclusion principle:
\[ 300 = 125 + 145 + 90 - (|A \cap B| + |B \cap C| + |A \cap C|) + x \]
- This simplifies to:
\[ 300 = 360 - (|A \cap B| + |B \cap C| + |A \cap C|) + x \]
- Rearranging gives:
\[ |A \cap B| + |B \cap C| + |A \cap C| = 60 + x \quad \text{(Equation 1)} \]

6. **Using the Second Equation:**
- From the second equation about students liking exactly two games:
\[ |A \cap B| + |B \cap C| + |A \cap C| - 3x = 32 \quad \text{(Equation 2)} \]

7. **Substituting Equation 1 into Equation 2:**
- Substitute \( |A \cap B| + |B \cap C| + |A \cap C| \) from Equation 1 into Equation 2:
\[ (60 + x) - 3x = 32 \]
- This simplifies to:
\[ 60 - 2x = 32 \]
- Rearranging gives:
\[ 2x = 60 - 32 = 28 \]
- Thus, we find:
\[ x = 14 \]

8. **Conclusion:**
- Therefore, the number of students who like all three games is \( 14 \).

### Final Answer:
The number of students who like to play all three games is **14**.