Please Help with explanation!!!, Question; Jeffrey works as a DJ at a local radio station.

Question

1.Jeffrey works as a DJ at a local radio station. On occasion, he chooses some of the songs he will play based on the phone-in requests received by the switchboard the previous day. Jeffrey’s list of 200 possible selections includes,

1. All the songs in the top 100
2. 134 hard-rock songs
3. 50 phone-in requests

• 45 hard-rock songs in the top 100

• 20 phone-in requests in the top 100

• 24 phone-in requests for hard-rock songs

AND yes I did a Venn Diagram

a) how many phone-in requests were for hard-rock songs in the top 100.

b) how many of the songs in the top 100 were neither phone-in requests nor hard-rock selections

Answer 1

Let

T be the set of top songs,

H be the set of hard-rock songs,

P be the set of songs requested by phone.

Using the notation

|S| to denote the size of a set S,

we are given:

|T| = 100

|H| = 134

|P| = 50

|T∩H| = 45

|T∩P| = 20

|H∩P| = 24

|T ∪ H ∪ P| = 200

a)

In your previous question about student registration I explained why

|T ∪ H ∪ P| = |T| + |H| + |P| - |T∩H| - |H∩P| - |T∩P| + |T∩H∩P|

From that

|T∩H∩P| = |T ∪ H ∪ P| - |T| - |H| - |P| + |T∩H| + |H∩P| + |T∩P| =

200 - 100 - 134 - 50 + 45 +20 + 24 = 5

b)

Given that you have drawn the Venn diagram you can probably see that

the answer is represented by

|T| - |T∩H| - |T∩P| + |T∩H∩P| = 100 - 24 - 20 + 5 = 61