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In a class of 40 students, 22 study art, 18 study biology and 6 study only chemistry. 5 study all 3 subjects, 9 study art and biology, 7 study art but not chemistry and 11 study exactly one subject. illustrate the info on a Venn diagram. how many students study exactly two subjects. how many students study none of the subjects. what is the probability that a student chosen at random study chemistry.

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  1. draw 3 intersecting circles, label them A, B, C, for the subject names
    fill in from the centre outwards
    you are told 5 take all three, so put 5 in the intersection of all 3
    Then do the "two at a time" sections of the Venn diagram
    e.g. 9 study art and chemistry. BUT, we already have 5 counted
    in that part, so put 4 in the part representing only art and biology.
    But 6 in the "only chemistry" part.

    We don't know the art and chemistry, but not biology, call that x
    Since we know art has 22, we know that the "only art" part would be
    22 - 4-4-x = 13-x

    We know that 7 study art but not chemistry
    13-x + 4 = 7
    x = 10

    replace the x with 10

    Label the "only chemistry and biology" part as y
    from our data:
    then the only biology part is
    18-4-5-y = 9-y

    we know 11 study exactly one subject
    13-x + 6 + 9-y = 11
    13-10 + 6 + 9 -y = 11
    y = 7

    replace all the y's with 7

    looks like all the part are filled in, but I only count 37, and in the class of
    40 students, 3 must not take any of the listed subjects.
    counting up the chemistry numbers, there are 28

    so the prob that a random student of the class selected takes chemistry
    = 28/40
    = 7/10 or 0.7

    check my arithmetic, the numbers seem to make sense

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