A survey was conducted among 300 students. It was found that 125 students like to 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 three games?

Options

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

Answer

To solve the problem step by step, we can use the principle of inclusion-exclusion. Let's denote:

- \( C \): the set of students who like cricket
- \( F \): the set of students who like football
- \( T \): the set of students who like tennis

### Step 1: Define the known values
From the problem, we know:
- Total number of students, \( n = 300 \)
- Students who like cricket, \( |C| = 125 \)
- Students who like football, \( |F| = 145 \)
- Students who like tennis, \( |T| = 90 \)
- Students who like exactly two games, \( |C \cap F| + |F \cap T| + |T \cap C| - 3|C \cap F \cap T| = 32 \)

### Step 2: Use the inclusion-exclusion principle
According to the principle of inclusion-exclusion, the total number of students who like at least one of the games can be given by:
\[ |C \cup F \cup T| = |C| + |F| + |T| - |C \cap F| - |F \cap T| - |T \cap C| + |C \cap F \cap T| \]
Substituting the known values, we have:
\[ 300 = 125 + 145 + 90 - (|C \cap F| + |F \cap T| + |T \cap C|) + |C \cap F \cap T| \]
This simplifies to:
\[ 300 = 360 - (|C \cap F| + |F \cap T| + |T \cap C|) + |C \cap F \cap T| \]
Rearranging gives:
\[ |C \cap F| + |F \cap T| + |T \cap C| - |C \cap F \cap T| = 60 \]

### Step 3: Set up the equations
From the previous step, we have:
1. \( |C \cap F| + |F \cap T| + |T \cap C| - |C \cap F \cap T| = 60 \) (Equation 1)
2. \( |C \cap F| + |F \cap T| + |T \cap C| - 3|C \cap F \cap T| = 32 \) (Equation 2)

### Step 4: Solve the equations
From Equation 1:
\[ |C \cap F| + |F \cap T| + |T \cap C| = 60 + |C \cap F \cap T| \]
Substituting this into Equation 2:
\[ 60 + |C \cap F \cap T| - 3|C \cap F \cap T| = 32 \]
This simplifies to:
\[ 60 - 2|C \cap F \cap T| = 32 \]
Rearranging gives:
\[ -2|C \cap F \cap T| = 32 - 60 \]
\[ -2|C \cap F \cap T| = -28 \]
Dividing by -2:
\[ |C \cap F \cap T| = 14 \]

### Step 5: Conclusion
Thus, the number of students who like all three games is \( \boxed{14} \).