3. The campus radio station WVAS surveyed 190 students to determine the types of music they liked. The survey revealed that 114 liked rock, 50 liked jazz, and 41 likes blues. Moreover, 14 liked rock and jazz, 15 liked rock and blues, 11 liked blues and jazz, and 5 liked all three types of music.
a. How many students like jazz only?
b. How many students like jazz but not rock?
c. How many students like blues and jazz, but not rock
d. How many students like exactly one of the three types of music?
e. How many students like at least two of the three types of music?
a,b,d,e
To solve this problem, we can use the principle of inclusion and exclusion. We'll calculate the number of students who like each type of music, and then subtract the overlaps to get the desired results.
a. To find the number of students who like jazz only, we need to exclude those who like jazz and another type of music. Therefore, we subtract the number of students who like both jazz and rock (14) and those who like both jazz and blues (11) from the total number of students who like jazz (50).
Number of students who like jazz only = Total number of students who like jazz - Number of students who like both jazz and rock - Number of students who like both jazz and blues
Number of students who like jazz only = 50 - 14 - 11 = 25
b. To find the number of students who like jazz but not rock, we just need to subtract the number of students who like both jazz and rock (14) from the number of students who like jazz only (found in part a).
Number of students who like jazz but not rock = Number of students who like jazz only - Number of students who like both jazz and rock
Number of students who like jazz but not rock = 25 - 14 = 11
c. To find the number of students who like blues and jazz, but not rock, we need to exclude those who like all three types of music (5) and those who like both blues, jazz, and rock (as this would include rock as well). Therefore, we subtract these overlaps from the number of students who like both blues and jazz (11).
Number of students who like blues and jazz, but not rock = Number of students who like both blues and jazz - Number of students who like all three types of music
Number of students who like blues and jazz, but not rock = 11 - 5 = 6
d. To find the number of students who like exactly one of the three types of music, we need to add up the number of students who like each type of music individually (rock, jazz, and blues), and then subtract the overlaps (those who like two or all three types of music).
Number of students who like exactly one of the three types of music = Number of students who like rock only + Number of students who like jazz only + Number of students who like blues only - Number of students who like two or more types of music
Number of students who like exactly one of the three types of music = (114 - 14 - 15) + (50 - 14 - 11) + (41 - 11 - 15) - 5
Number of students who like exactly one of the three types of music = 85
e. To find the number of students who like at least two of the three types of music, we need to add up the number of students who like two types of music and those who like all three types.
Number of students who like at least two of the three types of music = Number of students who like both rock and jazz + Number of students who like both rock and blues + Number of students who like both blues and jazz + Number of students who like all three types of music
Number of students who like at least two of the three types of music = 14 + 15 + 11 + 5
Number of students who like at least two of the three types of music = 45
Therefore:
a. 25 students like jazz only.
b. 11 students like jazz but not rock.
c. 6 students like blues and jazz, but not rock.
d. 85 students like exactly one of the three types of music.
e. 45 students like at least two of the three types of music.