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Ket [755]
3 years ago
14

The 65 students in a classical music lecture class were polled, with the following results: 37 like Wolfgang Amadeus Mozart 36 l

ike Ludwig van Beethoven 31 like Franz Joseph Haydn 14 like Mozart and Beethoven 21 like Mozart and Haydn 14 like Beethoven and Haydn 8 like all three composers. How many of these students like:________.
a. exactly two of these composers?
b. exactly one of these composers?
c. like only Mozart?
d. like Beethoven and Haydn, but not Beethoven?
e. like none of these composers?
Mathematics
1 answer:
Anna35 [415]3 years ago
4 0

Answer:

a) 25

b) 30

c) 10

d) Not Mozart, 6

e) 2

Step-by-step explanation:

We use a Venn Diagram to solve this question.

I am going to say that:

A are the students who like Mozart.

B are the students who like Beethoven

C are the students who like Haydn.

We have that:

A = a + (A \cap B) + (A \cap C) + (A \cap B \cap C)

In which a are those who only like Mozart, (A \cap B) are those who like Mozart and Beethoven, (A \cap C) are those who like Mozart and Haydn and (A \cap B \cap C) are those who like all three of them.

By the same logic, we have that:

B = b + (A \cap B) + (B \cap C) + (A \cap B \cap C)

C = c + (B \cap C) + (A \cap C) + (A \cap B \cap C)

We start finding these values from the intersection:

8 like all three composers

This means that A \cap B \cap C = 8

14 like Beethoven and Haydn

This means that:

(B \cap C) + (A \cap B \cap C) = 14

So

B \cap C = 6

21 like Mozart and Haydn

This means that:

(A \cap C) + (A \cap B \cap C) = 21

Then

A \cap C = 13

14 like Mozart and Beethoven

This means that:

(A \cap B) + (A \cap B \cap C) = 14

A \cap B = 6

31 like Franz Joseph Haydn

This means that C = 31. So

C = c + (B \cap C) + (A \cap C) + (A \cap B \cap C)

31 = c + 6 + 13 + 8

c = 4

36 like Ludwig van Beethoven

This means that B = 36

So

B = b + (A \cap B) + (B \cap C) + (A \cap B \cap C)

36 = b + 6 + 6 + 8

b = 16

37 like Wolfgang Amadeus Mozart

This means that A = 37. Then

A = a + (A \cap B) + (A \cap C) + (A \cap B \cap C)

37 = a + 6 + 13 + 8

a = 10

a. exactly two of these composers?

(A \cap B) + (A \cap C) + (B \cap C) = 6 + 13 + 6 = 25

b. exactly one of these composers?

a + b + c = 10 + 16 + 4 = 30

c. like only Mozart?

a = 10

d. like Beethoven and Haydn, but not Beethoven?

I will use not Mozart.

So B \cap C = 6

Not Mozart, 6.

e. like none of these composers?

At least 1:

(A \cup B \cup C) = a + b + c + (A \cap B) + (A \cap C) + (B \cap C) + (A \cap B \cap C) = 10 + 16 + 4 + 6 + 13 + 6 + 8 = 63

The total is 65

So 65 - 63 = 2 like none of these composers

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