Agroup of students were surveyed about their taste in music. they were asked how they like three types of music: rock, country western and jazz. the results are summarized below no one dislikes all three. six like all three. eleven don't like country western. sixteen like jazz. 3 like country western and jazz but not rock. 6 don't like rock. 8 like rock and country western. 7 don't like jazz. how many students were surveyed?

23

Step-by-step explanation:

This problem can be solved using a Venn diagram (see picture attached).

Notice how the data was included in the diagram. Data:

Six like all three (the intersection of all 3 circles)Eleven don't like country western. This means that a + b + c = 11 (eq. 1)Sixteen like jazz. This means that 3 + 6 + b + c = 16 or b + c = 7 (eq. 2)3 like country western and jazz but not rock (the intersection of only  country western and jazz)6 don't like rock. This means that d + 3 + c  = 6 (eq. 3)8 like rock and country western. This means that the intersection of only rock and country western without jazz is 27 don't like jazz. This means that a + 2 + d = 7 (eq. 4)

Combining equation 1 with equation 2 we get:

a + 7 = 11 or a = 4

Combining this result and eq. 4:

4 + 2 + d = 7 or d = 1

Combining this result and eq. 3:

1 + 3 + c = 6 or c = 2

Combining this result and eq. 2:

b + 2 = 7 or b = 5

There were surveyed a + b + c + d + 2 + 6 + 3 or 4 + 5 + 2 + 1 + 2 + 6 + 3 = 23 students.