Question

Help me solve please 5 I’ve attached questions A-D that need to be answered as well

Answer 1

Given:

The number of students like Wolfgang Amadeus Mozart = 36

The number of students like Ludwig von Beethoven = 35

The number of students like Franz Joseph Hayden = 30

The number of students like Mozart and Beethoven = 14

The number of students like Mozart and Hayden = 21.

The number of students like Beethoven and Hayden = 14.

The number of students like all three composers = 9

Required:

We need to draw a ven diagram for the given information.

Explanation:

Let M be the set of students that like Mozart, B be the set of students that like Beethoven, and H be the set of students that like Hayden.

a)

We need to find the number of students like exactly two of these composers.

We know that

M and B =14

B and H =14

M and H =21

$n(M\cap B)=14,\text{ }n(B\cap H)=14,\text{ and }n(M\cap H)=21$

The number of students like exactly two of these composers.

$n(M\cap B)+n(B\cap H)+n(M\cap H)=14+14+21=49$

49 students like exactly two of these composers s

b)

We need to find the number of students like exactly one of these composers.

We know that

$n(M)=36,\text{ }n(B)=35,\text{ }n(H)=30,\text{ n}(M\operatorname{\cap}B)=14,\text{ n}(B\operatorname{\cap}H)=14,n(M\operatorname{\cap}H)=21,\text{ and }n(M\cap H\cap B)=9.$

$n(M\cup B\cup H)=m(M)+\text{ }n(B)+n(H)-n(M\operatorname{\cap}B)-\text{n}(B\operatorname{\cap}H)-n(M\operatorname{\cap}H)+n(M\cap H\cap B)$$n(M\cup B\cup H)=36+35+30-14-14-21+9.$$n(M\cup B\cup H)=61$

The number of students like the composers =61.

c)

We need to find the number of students who do not like these composers.

We know that

$n(M)=36,\text{ }n(B)=35,\text{ }n(H)=30,\text{ n}(M\operatorname{\cap}B)=14,\text{ n}(B\operatorname{\cap}H)=14,n(M\operatorname{\cap}H)=21,\text{ and }n(M\cap H\cap B)=9.$

$n(M\cup B\cup H)=m(M)+\text{ }n(B)+n(H)-n(M\operatorname{\cap}B)-\text{n}(B\operatorname{\cap}H)-n(M\operatorname{\cap}H)+n(M\cap H\cap B)$$n(M\cup B\cup H)=36+35+30-14-14-21+9.$$n(M\cup B\cup H)=61$

The number of students like the composers =61.

The total number of students = 64

The number of students who do not like composers=64-61 =3

3 students like none of the composers.

d)

We need to find the number of students who like Mozert but neither like Beethoven and Hayden.

$n(M)-n(M\cap B)-n(M\cap H)+n(M\cap H\cap B)$

Substitute know values.

$n(M)=36,\text{ n}(M\operatorname{\cap}B)=14,n(M\operatorname{\cap}H)=21,\text{ and }n(M\cap H\cap B)=9.$$=36-14-21+9=10$

10 students who like Mozert but neither like Beethoven and Hayden.

Final answer:

a) 21

b) 61

c) 3

d) 10