Use the following Information to determine the number of peo

Use the following Information to determine the number of people in each region of the Venn Diagram 190 people were asked which of these instruments that they could play: Piano, Drums, or Guitar. 44 people could play none of these instruments. 21 people could play all three of these instruments. 59 people could play drums or guitar but could not play piano. 70 people could play guitar. 86 people could play at least two of these instruments. 11

Solution

Based on the figure given we can define the variables as shown below,

a=people playing only Guitar

b=people playing Guiter or Piano but not Drums

c=people playing only Piano

d=people playing Guiter or Drums but not Piano

e=people playing all three of these

f=people playing Drums and Piano but not Guitar

g=people playing only Drums

h=people playing none of these.

So, we have, h=44, e=21, b=11

also, given that 70 people could play Guitar so, we have

a+b+d+e=70 a+11+d+21=70 a+d=38

86 people could play at least two these instruments, so we have,

b+d+e+f=86 11+d+21+f=86 d+f=54

52 peoply could play piano and drums, so we have,

e+f=52 21+f=52 f=31

59 people could play drums or guitar but could not play piano, so we have,

a+d+g=59 38+g=59, Since (a+d=38)

g=59-38=21

Then, f=31 and d+f=54 implies d=54-31=23

and a+d=38 and d=23 implies a=38-23=15

People playing at one game=total number of people-people not playing all the games

a+b+c+d+e+f+g=190-h

15+11+c+23+21+31+21=190-44

c+122=146

c=146-122

c=24

Therefore, we have, a=15, b=11, c=24 ,d=23,e=21, f=31, g=21, h=44