Because 12 provided all three features, this means that the number 12 goes in the very center of the three circles. Now since "21 provided laundry services and business centers", this means that 21-12=9 provided laundry and business centers (remember, some of the 21 might have provided all three)
Also, since "29 provided refrigerators in the guest rooms and business centers", this means that 29-12=17 offered refrigerators and business centers only. In addition, because " 23 provided refrigerators in the guest rooms and laundry services ", this means that 23-12=11 offered refrigerators and laundry only.
So place the value 9 in the region that overlaps the upper circles. Place the value 17 in the region between the circles labeled "refrigerators" and "business centers". Finally, place the value 11 in the region between " laundry " and "refrigerators"
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Now we're given that 37 hotels offer laundry. Now looking at the venn diagram, add up the values that lie in the "laundry" circle to get: 9+12+11=32
Now subtract this value from 37 to get: 37-32=5. So there are 13 hotels who ONLY provide laundry.
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Also, we're told that 44 provided business centers. Now add up all of the values that lie within the "business center" circle to get: 9+12+17=38
Subtract this result from 44 to get: 44-38=6. So there are 6 hotels who ONLY offer business centers.
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In addition, it's stated that "45 provided refrigerators in the guest rooms". Add up the values that reside within the "refrigerator"s circle to get: 11+12+17=40
Subtract this value from 45 to get: 45-40=5. So there are 5 hotels who ONLY offer refrigerators.
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Finally, add up EVERY value that has been written in the circles to get: 5+9+6+11+12+17+5=65
So there are 65 hotels who provide at least one service listed (ie they may offer one, two or all three features). Now subtract this result from 75 (the total given) to get 75-65=10
So there are 10 hotels who do not offer any of the services listed
Now let's use the venn diagram to answer the questions
a) How many of the resorts provided only refrigerators in the guest rooms? 5 Hotels
b) How many of the resorts provided exactly one of the features? Just add the values that pertain to the hotels who offer only one service: 5+5+6=16. So there are 16 resorts who offer exactly one feature
c) How many of the resorts provided at least one of the features? There are many ways to do this, but take note that there are 10 who do NOT offer any of the features. So subtract 10 from 75 to get 65. So there are 65 hotels who offer at least one feature
d) How many of the resorts provided exactly two of the features? Simply add up the values that lie in between each circle (don't add in the value in the very center) to get: 9+11+17=37. So there are 37 hotels who offer exactly two services
e) How many of the resorts provided none of the features? The value that lies outside every circle is the number of hotels which offered none of the features. So 10 hotels did not offer any feature listed.