There are 45 Musicians in an orchestra , and all play two instruments . Of these musicians , 36 play the piano , and 22 play the violin . What is the maximum possible number of orchestra members who play both the piano and the violin ?

Options

A. 9
B. 13
C. 22
D. 23

Answer

To find the maximum possible number of orchestra members who play both the piano and the violin, we can use the principle of inclusion-exclusion. Here's how we can solve the problem step by step:

### Step 1: Define the Variables
Let:
- \( P \) = number of musicians who play the piano = 36
- \( V \) = number of musicians who play the violin = 22
- \( N \) = total number of musicians in the orchestra = 45
- \( x \) = number of musicians who play both instruments

### Step 2: Use the Inclusion-Exclusion Principle
According to the inclusion-exclusion principle, the total number of musicians can be expressed as:
\[ N = P + V - x \]
Where \( x \) is the number of musicians who play both instruments.

### Step 3: Substitute the Known Values
Substituting the known values into the equation:
\[ 45 = 36 + 22 - x \]

### Step 4: Simplify the Equation
Combine the numbers on the right side:
\[ 45 = 58 - x \]

### Step 5: Solve for \( x \)
Rearranging the equation to solve for \( x \):
\[ x = 58 - 45 \]
\[ x = 13 \]

### Conclusion
The maximum possible number of orchestra members who play both the piano and the violin is \( 13 \).