Assume that 140 surveys are completed. Of those surveyed, 71 responded positively to effectiveness, 60 responded positively to side effects, and 65 responded positively to cost. Also, 33 responded positively to both effectiveness and side effects, 31 to effectiveness and cost, 28 to side effects and cost, and 21 to none of the items. How many responded positively to all three?

15

Step-by-step explanation:

Let n (T) denotes total surveys done i. e. n (T) = 140

Let n (A) be the no. of responses to positively to effectiveness i. e. n (A) = 71

Let n (B) be the no. of side effects i. e. n (B) = 60

Let n (C) be the no. of responses to cost i. e. n (C) = 65

33 responded positively to both effectiveness and side effects

So, n (A∩B) = 33

31 to effectiveness and cost

n (A∩C) = 31

28 to side effects and cost

n (B∩C) = 28

21 to none of the items

So, n (A∪B∪C) = 140-21 = 119

we are supposed to find ow many responded positively to all three i. e. n (A∩B∩C)

Formula:

n (A∪B∪C) = n (A) + n (B) + n (C) - n (A∩B) - n (A∩C) - n (B∩C) + n (A∪B∪C)

119=71+60+65-33-31-28 + n (A∪B∪C)

119=104 + n (A∪B∪C)

119-104 = n (A∪B∪C)

15 = n (A∪B∪C)

Hence 15 responded positively to all three