In a class of 35 students, 24 students like cricket and 16 like football. If each student plays atleast one game, find how many students like both games ?

Answer

To solve the problem step by step, we can use the principle of inclusion-exclusion. Here’s how we can find the number of students who like both cricket and football.

### Step-by-Step Solution:

1. **Identify the given data:**
- Total number of students (N) = 35
- Number of students who like cricket (C) = 24
- Number of students who like football (F) = 16

2. **Set up the formula for union of two sets:**
The formula for the union of two sets is:
\[ N(C \cup F) = N(C) + N(F) - N(C \cap F) \]
where \(N(C \cup F)\) is the total number of students who like either cricket or football, \(N(C)\) is the number of students who like cricket, \(N(F)\) is the number of students who like football, and \(N(C \cap F)\) is the number of students who like both games.

3. **Substitute the known values into the formula:**
We know:
\[ N(C \cup F) = 35, \quad N(C) = 24, \quad N(F) = 16 \]
Plugging these values into the formula gives:
\[ 35 = 24 + 16 - N(C \cap F) \]

4. **Simplify the equation:**
Combine the numbers on the right side:
\[ 35 = 40 - N(C \cap F) \]

5. **Isolate \(N(C \cap F)\):**
Rearranging the equation to solve for \(N(C \cap F)\):
\[ N(C \cap F) = 40 - 35 \]
\[ N(C \cap F) = 5 \]

6. **Conclusion:**
Therefore, the number of students who like both cricket and football is **5**.