100 students were asked to fill 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? [a] 2. How many students checked exactly two boxes? [b] 3. How many students checked at LEAST two boxes? [c]4. How many students checked the Clubs box only?

1. 45

2. 9

3. 11

4. 17

Explanation:

Given the following:

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.

Hence N (H) = 28

N (C) = 26

N (D) = 14

N (H U C) = 8

N (H U D) = 4

N (C U D) = 3

N (H U C U D) = 2

We also know that

Total = N (H) + N (C) + N (D) - N (H U C) - N (H U D) - N (C U D) + N (H U C U D)

Substituting the given values, we obtain

Total = 55

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

2. Students who checked exactly two box

= N (H U C) + N (H U D) + N (C U D) - 3N (H U C U D) (from probability theorem)

Substituting the values, we have 8 + 4 + 3 - 6 = 9 students

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

N (H U C) + N (H U D) + N (C U D) - 2N (H U C U D) = 8 + 4 + 3 - 4 = 11 students

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 students.