In a group of students, 225 students know French, 100 know Spanish and 45 know both. Each student knows either French or Spanish. How many students are there in the group?

Answer

To solve the problem step by step, we can use the principle of inclusion-exclusion in set theory.

### Step 1: Identify the given information
- Let \( F \) be the set of students who know French.
- Let \( S \) be the set of students who know Spanish.
- We know that:
- \( |F| = 225 \) (students know French)
- \( |S| = 100 \) (students know Spanish)
- \( |F \cap S| = 45 \) (students know both French and Spanish)

### Step 2: Use the principle of inclusion-exclusion
The total number of students who know either French or Spanish can be calculated using the formula:
\[ |F \cup S| = |F| + |S| - |F \cap S| \]
where \( |F \cup S| \) is the number of students who know either French or Spanish.

### Step 3: Substitute the values into the formula
Substituting the values we have:
\[ |F \cup S| = 225 + 100 - 45 \]

### Step 4: Calculate the total
Now, perform the calculation:
\[ |F \cup S| = 325 - 45 = 280 \]

### Step 5: Conclusion
Thus, the total number of students in the group is \( 280 \).

### Final Answer
The total number of students in the group is **280**.

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