Question:
In a group of 25, 10 have studied French, 11 have studied Spanish, and 8 have not studied either. How many of these studied both French and Spanish?
In a group of 25, 10 have studied French, 11 have studied Spanish, and 8 have not studied either. How many of these studied both French and Spanish?
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Drawing a Venn diagram helps visualize the problem instantly. The sum of the individual groups plus the "neither" group minus the intersection must equal the total population: \(10 + 11 + 8 - x = 25 \Rightarrow 29 - x = 25 \Rightarrow x = 4\).
Updated On: May 9, 2026
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The Correct Option is B
Solution and Explanation
Step 1: Understanding the Question:
This is a standard set theory problem. We are given the total number of people, the number of people in two individual sets, and the number of people outside both sets. We need to find the intersection of the two sets.
Step 2: Key Formula or Approach:
Use the principle of inclusion-exclusion for two sets: \[ n(A \cup B) = n(A) + n(B) - n(A \cap B) \] Where \(n(A \cup B)\) is the number of people who studied at least one of the subjects, which can be found by subtracting the "neither" group from the total group.
Step 3: Detailed Explanation:
Let \(F\) be the set of people who studied French, and \(S\) be the set of people who studied Spanish.
Total number of people in the group \(= 25\).
Number of people who studied neither \(= 8\).
Number of people who studied at least one language \(n(F \cup S) = 25 - 8 = 17\).
We are given: \(n(F) = 10\)
\(n(S) = 11\)
Substitute these values into the inclusion-exclusion formula: \[ 17 = 10 + 11 - n(F \cap S) \] \[ 17 = 21 - n(F \cap S) \] \[ n(F \cap S) = 21 - 17 = 4 \] Therefore, \(4\) people studied both French and Spanish.
Step 4: Final Answer:
The number of people who studied both is \(4\).
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