Questions: A survey of 114 college students was taken to determine the musical styles they liked. Of those, 39 students listened to rock, 50 to classical, and 36 to jazz. Also, 11 students listened to rock and jazz, 25 to rock and classical, and 20 to classical and jazz. Finally, 9 students listened to all three musical styles. Construct a Venn diagram and determine the cardinality for each region. How many students listened to classical and jazz, but not rock? n(classical and jazz, not rock) =

A survey of 114 college students was taken to determine the musical styles they liked. Of those, 39 students listened to rock, 50 to classical, and 36 to jazz. Also, 11 students listened to rock and jazz, 25 to rock and classical, and 20 to classical and jazz. Finally, 9 students listened to all three musical styles. Construct a Venn diagram and determine the cardinality for each region. How many students listened to classical and jazz, but not rock?

n(classical and jazz, not rock) =

Transcript text: Points: 0 of 1 Save A survey of 114 college students was taken to determine the musical styles they liked. Of those, 39 students listened to rock, 50 to classical, and 36 to jazz. Also, 11 students listened to rock and jazz, 25 to rock and classical, and 20 to classical and jazz. Finally, 9 students listened to all three musical styles. Construct a Venn diagram and determine the cardinality for each region. How many students listened to classical and jazz, but not rock? $\mathrm{n}($ classical and jazz, not rock $)=$ $\square$

Solution

Solution Steps

To solve this problem, we need to use the principle of inclusion-exclusion to determine the number of students who listened to classical and jazz but not rock. We will calculate the number of students who listened to both classical and jazz, and then subtract those who also listened to rock.

Solution Approach

Calculate the total number of students who listened to both classical and jazz.
Subtract the number of students who listened to all three musical styles from this total to get the number of students who listened to classical and jazz but not rock.

Step 1: Given Data

We have the following information from the survey of 114 college students:

$ n(\text{rock}) = 39 $
$ n(\text{classical}) = 50 $
$ n(\text{jazz}) = 36 $
$ n(\text{rock and jazz}) = 11 $
$ n(\text{rock and classical}) = 25 $
$ n(\text{classical and jazz}) = 20 $
$ n(\text{all three}) = 9 $

Step 2: Calculate Students Listening to Classical and Jazz but Not Rock

To find the number of students who listened to classical and jazz but not rock, we use the formula: \[ n(\text{classical and jazz, not rock}) = n(\text{classical and jazz}) - n(\text{all three}) \] Substituting the values: \[ n(\text{classical and jazz, not rock}) = 20 - 9 = 11 \]