Question

Out of 400 students who attended a seminar, 240 opted for written exam, 200 opted for presentations, 160 opted for assignments and 20 did not opt for giving any of the three methods of test. 200 students had exactly one of the three methods of test. How many students opted for exactly two of the three testing methods?

• A. 140
• B. 160
• C. 220
• D. 170

Answer 1

The Correct Option is A

This logical reasoning question involves understanding the distribution of students based on their choices of test methods. Let's break down the problem step by step:

1. Total number of students who attended the seminar = 400.
2. From these, 20 students did not opt for any of the three methods. Thus, the students who opted for at least one method = \(400 - 20 = 380\).
3. Let's denote:
   • N(W) = 240 for students who opted for the written exam.
   • N(P) = 200 for students who opted for presentations.
   • N(A) = 160 for students who opted for assignments.
4. From the given data, 200 students opted for exactly one method.

We are required to find how many students opted for exactly two out of the three testing methods. Let’s denote this as x. Also, let's find N(W \cap P \cap A) as the number of students who opted for all three methods.

5. The sum of students opting for exactly one method, exactly two methods, and all three methods should equal 380:
   \[ 200 + x + N(W \cap P \cap A) = 380 \implies x + N(W \cap P \cap A) = 180 \]
6. By applying the principle of inclusion-exclusion for three sets:
   \[ N(W) + N(P) + N(A) - (N(W \cap P) + N(W \cap A) + N(P \cap A)) + N(W \cap P \cap A) = 380 \]
   \[ 240 + 200 + 160 - (N(W \cap P) + N(W \cap A) + N(P \cap A)) + N(W \cap P \cap A) = 380 \]
7. Simplifying gives:
   \[ 600 - (N(W \cap P) + N(W \cap A) + N(P \cap A)) + N(W \cap P \cap A) = 380 \]
   \[ N(W \cap P) + N(W \cap A) + N(P \cap A) - N(W \cap P \cap A) = 220 \]
8. From equations derived:
   \[ x = 220 - N(W \cap P \cap A) \]

From steps 5 and 8:

\[ x + N(W \cap P \cap A) = 180 \\ x = 220 - N(W \cap P \cap A) \]

10. Solving:
    \[ x = 220 - (220 - x) \\ 2x = 180 \\ x = 140 \]

Therefore, the number of students who opted for exactly two of the three testing methods is 140.