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If in a certain school 20 students are t...

If in a certain school 20 students are taking math and 10 are taking history and 7 are taking both, how many students are taking either math or history?

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To solve the problem of how many students are taking either math or history, we can use the principle of inclusion-exclusion. ### Step-by-Step Solution: 1. **Identify the given numbers:** - Let \( n(A) \) be the number of students taking math = 20 - Let \( n(B) \) be the number of students taking history = 10 - Let \( n(A \cap B) \) be the number of students taking both math and history = 7 2. **Use the formula for the union of two sets:** The formula to find the number of students taking either math or history is given by: \[ n(A \cup B) = n(A) + n(B) - n(A \cap B) \] where \( n(A \cup B) \) is the number of students taking either math or history. 3. **Substitute the values into the formula:** \[ n(A \cup B) = 20 + 10 - 7 \] 4. **Calculate the result:** - First, add the number of students taking math and history: \[ 20 + 10 = 30 \] - Then, subtract the number of students taking both subjects: \[ 30 - 7 = 23 \] 5. **Conclusion:** Therefore, the number of students taking either math or history is \( 23 \). ### Final Answer: 23 students are taking either math or history.
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