Participation in sports is compulsory in a school and A class has `80` student out of which `60` students play football and `40` subjects play basketball. Find how many students.
`(a) ` play football only,
`(b)` play basketball only,
`(c )` Play both football and basketball.

Answer

To solve the problem, we will use the principle of set theory and Venn diagrams to find the number of students participating in football only, basketball only, and both sports.

### Step-by-Step Solution:

1. **Identify the Given Information:**
- Total number of students (n) = 80
- Students playing football (n(F)) = 60
- Students playing basketball (n(B)) = 40

2. **Use the Formula for Union of Two Sets:**
The formula for the union of two sets is given by:
\[ n(F \cup B) = n(F) + n(B) - n(F \cap B) \]
Where:
- \( n(F \cup B) \) is the total number of students playing either football or basketball.
- \( n(F \cap B) \) is the number of students playing both football and basketball.

3. **Substitute the Known Values:**
We know that \( n(F \cup B) = 80 \), \( n(F) = 60 \), and \( n(B) = 40 \). Plugging these values into the formula gives:
\[ 80 = 60 + 40 - n(F \cap B) \]

4. **Solve for \( n(F \cap B) \):**
Rearranging the equation:
\[ 80 = 100 - n(F \cap B) \]
\[ n(F \cap B) = 100 - 80 = 20 \]
Thus, the number of students who play both football and basketball is 20.

5. **Calculate Students Playing Football Only:**
To find the number of students who play football only, we subtract the number of students who play both sports from the total number of football players:
\[ n(\text{Football only}) = n(F) - n(F \cap B) = 60 - 20 = 40 \]

6. **Calculate Students Playing Basketball Only:**
Similarly, to find the number of students who play basketball only, we subtract the number of students who play both sports from the total number of basketball players:
\[ n(\text{Basketball only}) = n(B) - n(F \cap B) = 40 - 20 = 20 \]

### Final Answers:
- (a) Students who play football only: **40**
- (b) Students who play basketball only: **20**
- (c) Students who play both football and basketball: **20**