In a group of 30 students, 20 take French, 15 take Spanish, and 5 take neither language. How many students take both French and Spanish?

Options

A. 0
B. 5
C. 10
D. 15

Explanation

To solve the problem, we will use the principle of inclusion-exclusion and set up a Venn diagram to visualize the relationships among the students taking French, Spanish, and neither language.

### Step-by-Step Solution:

1. **Identify the total number of students and the given data:**
- Total students = 30
- Students taking French (F) = 20
- Students taking Spanish (S) = 15
- Students taking neither language = 5

2. **Define the variables:**
- Let \( C \) be the number of students taking both French and Spanish.
- Let \( B \) be the number of students taking only French.
- Let \( D \) be the number of students taking only Spanish.
- Let \( A \) be the number of students taking neither language (which is given as 5).

3. **Set up the equations based on the information:**
- The total number of students can be expressed as:
\[ A + B + C + D = 30 \]
- The students taking French can be expressed as:
\[ B + C = 20 \]
- The students taking Spanish can be expressed as:
\[ C + D = 15 \]

4. **Substitute the known value of \( A \):**
- Since \( A = 5 \), we can substitute this into the first equation:
\[ 5 + B + C + D = 30 \]
- Simplifying this gives:
\[ B + C + D = 25 \quad \text{(Equation 1)} \]

5. **Use the second equation:**
- From \( B + C = 20 \) (Equation 2).

6. **Use the third equation:**
- From \( C + D = 15 \) (Equation 3).

7. **Now we have two equations (Equation 1 and Equation 2):**
- From Equation 1:
\[ B + C + D = 25 \]
- From Equation 2:
\[ B + C = 20 \]

8. **Subtract Equation 2 from Equation 1:**
\[ (B + C + D) - (B + C) = 25 - 20 \]
This simplifies to:
\[ D = 5 \]

9. **Substitute \( D \) back into Equation 3:**
- From \( C + D = 15 \):
\[ C + 5 = 15 \]
- Solving for \( C \):
\[ C = 15 - 5 = 10 \]

10. **Conclusion:**
- The number of students who take both French and Spanish is \( C = 10 \).

### Final Answer:
The number of students taking both French and Spanish is **10**.