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

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:

In which a are those who only like Mozart, are those who like Mozart and Beethoven, are those who like Mozart and Haydn and are those who like all three of them.

By the same logic, we have that:

We start finding these values from the intersection:

8 like all three composers

This means that

14 like Beethoven and Haydn

This means that:

So

21 like Mozart and Haydn

This means that:

Then

14 like Mozart and Beethoven

This means that:

31 like Franz Joseph Haydn

This means that C = 31. So

36 like Ludwig van Beethoven

This means that

So

37 like Wolfgang Amadeus Mozart

This means that A = 37. Then

a. exactly two of these composers?

b. exactly one of these composers?

c. like only Mozart?

d. like Beethoven and Haydn, but not Beethoven?

I will use not Mozart.

So

Not Mozart, 6.

e. like none of these composers?

At least 1:

The total is 65

So 65 - 63 = 2 like none of these composers

43

step-by-step explanation: