Question

100 students were asked to ll out a form with three survey questions, as follows: H: Honor Roll
C: Club membership (Robotics Club or Gaming Club)
D: Double-major

Survey results were as follows:

28 checked H (possibly non-exclusively), 26 checked C (possibly non-exclusively), 14 checked D (possibly non-exclusively)

8 checked H and C (possibly. non-exclusively), 4 checked H and D (possibly. non- exclusively), 3 checked C and D (possibly. non-exclusively)

And 2 checked all three statements.
1. How many students didn't check any of the boxes?

2. How many students checked exactly two boxes?
3. How many students checked at LEAST two boxes?

4. How many students checked the Clubs box only? [d]

Answer 1

Given

28 checked H

26 checked C

14 checked D

8 checked H and C

4 checked H and D

3 checked C and D

2 checked all.

We know

1) Students who didn't check any box = 100-55 = 45 students.

2) Students who checked exactly two box =

The people who have checked all three will be calculated once each in HUC, HUD and CUD, so they had to be subtracted thrice in HUCUD so as to not include them in exactly two box.

3) Students who checked atleast two box =

The people who have checked all three are needed to be calculated once. Earlier, we subtracted them thrice so we add one time

  

4) Given N(C) = 26

We subtract N(CUD) and N(HUC) as they have checked another apart from club.

   26-8-3=15

Now we could notice we have subtracted N(HUCUD) twice in both categories, so we add one time to neutralise

   15+2=17

Hence N(only C)=17.

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