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16 July, 19:54

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?

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  1. 16 July, 21:21
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    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.
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