In a class of 35 students, 24 like to play cricket and 16 like to play football. Also, each student likes to play at least one of the two games. How many students like to play both cricket and football ?

Answer

To find the number of students who like to play both cricket and football, we can use the principle of inclusion-exclusion from set theory. Let's break down the solution step by step.

### Step 1: Define the given values
- Total number of students (n) = 35
- Number of students who like cricket (n(A)) = 24
- Number of students who like football (n(B)) = 16

### Step 2: Use the principle of inclusion-exclusion
According to the principle of inclusion-exclusion, the total number of students who like at least one of the two games can be expressed as:
\[ n(A \cup B) = n(A) + n(B) - n(A \cap B) \]
Where:
- \( n(A \cup B) \) is the number of students who like either cricket or football or both.
- \( n(A \cap B) \) is the number of students who like both cricket and football.

### Step 3: Substitute the known values into the equation
Since every student likes at least one of the two games, we have:
\[ n(A \cup B) = 35 \]
Substituting the known values into the equation:
\[ 35 = 24 + 16 - n(A \cap B) \]

### Step 4: Simplify the equation
Combine the numbers on the right side:
\[ 35 = 40 - n(A \cap B) \]

### Step 5: Solve for \( n(A \cap B) \)
Rearranging the equation gives:
\[ n(A \cap B) = 40 - 35 \]
\[ n(A \cap B) = 5 \]

### Conclusion
Therefore, the number of students who like to play both cricket and football is **5**.

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