QuestionDetermine the number of students in a classical music lecture class who like different composers using a Venn diagram. Let M = Mozart, B = Beethoven, H = Haydn. Find the number of students who like: (a) exactly 2 composers, (b) exactly 1 composer, (c) none of the composers, (d) M but not B or H.

Studdy Solution
To find the number of students who like Mozart but neither Beethoven nor Haydn, use the number of students who like exactly one composer from Mozart's set.
$$|M|_{\text{only}} = \text{Number of students who like Mozart but neither Beethoven nor Haydn}$$
Now let's perform the calculations:
STEP_2: $$|M|_{\text{only}} = 38 - 15 - 21 + 9 = 11$$
STEP_3: $$|B|_{\text{only}} = 38 - 15 - 16 + 9 = 16$$
STEP_4: $$|H|_{\text{only}} = 33 - 21 - 16 + 9 = 5$$
STEP_6: $$|MB|_{\text{only}} = 15 - 9 = 6$$
STEP_7: $$|MH|_{\text{only}} = 21 - 9 = 12$$
STEP_8: $$|BH|_{\text{only}} = 16 - 9 = 7$$
STEP_11: $$\text{None} = 68 - (11 + 16 + 5 + 6 + 12 + 7 + 9) = 68 - 66 = 2$$
The answers to the questions are: (a) The number of students who like exactly two composers is $|MB|_{\text{only}} + |MH|_{\text{only}} + |BH|_{\text{only}} = 6 + 12 + 7 = 25$. (b) The number of students who like exactly one composer is $|M|_{\text{only}} + |B|_{\text{only}} + |H|_{\text{only}} = 11 + 16 + 5 = 32$. (c) The number of students who like none of the composers is $2$. (d) The number of students who like Mozart, but neither Beethoven nor Haydn is $|M|_{\text{only}} = 11$.