A class adviser surveys 35 of his students on the kind of pets they have at home. he found that 15 have dogs, 12 have cats and 8 have birds. five have dogs and cats, 4 have dogs and birds and 2 have cats and birds. if no one has all three kinds of pets how many students have none of these pets? please give a solution



I used a venn diagram to represent it.
Since no one has all three kinds, you start with putting who has 2 of different kinds (like place 4 between dogs and birds, etc) 
When you're done with putting who has dogs AND cats, dogs AND birds, cats AND birds, then you subtract that from the total of each kind.

Example: If 15 total have dogs, then that includes the ones who also have dogs and birds or dogs and cats. Since 5 have dogs and cats and 4 have dogs and birds, then that's 15-9. So you have 6 who only have dogs.

Add all the numbers in the venn diagram (=24) 

Since the total of people asked was 35 (your universal set) then the ones who don't have any pets is 35-24= 11.

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We let the number of people with dogs be D, cats be C, and birds be B. We let the no. of people with none of this be x.

We need to know that:
B + C + D - B∩C - B∩D - C∩D + B∩C∩D + x = 35
8 + 12 + 15 - 2 - 4 - 5 + 0 + x = 35
35 - 11 + x = 35
x - 11 = 0
x = 11