After polling a class of 20 music students by a show of hands, you find that 8 students play the guitar and 9 students play the piano. Given that information, what is the minimum number of students in this music class who play both the guiter and the piano?

Options

A. 0
B. 1
C. 8
D. 17

Answer

To find the minimum number of students who play both the guitar and the piano, we can use the principle of inclusion-exclusion.

1. **Identify the total number of students and those who play each instrument:**
- Total number of students (n) = 20
- Number of students who play guitar (G) = 8
- Number of students who play piano (P) = 9

2. **Set up the equation for the total number of students:**
The total number of students can be expressed as:
\[ n = G + P - B \]
where \( B \) is the number of students who play both instruments.

3. **Substitute the known values into the equation:**
\[ 20 = 8 + 9 - B \]
Simplifying this gives:
\[ 20 = 17 - B \]

4. **Rearranging the equation to solve for B:**
\[ B = 17 - 20 \]
\[ B = -3 \]
Since the number of students cannot be negative, this indicates that there are no students who play both instruments.

5. **Conclusion:**
The minimum number of students who play both the guitar and the piano is 0.