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?

Answer:15Step-by-step explanation:Let n(T) denotes total surveys done i.e. n(T)=140Let n(A) be the no. of responses to positively to effectiveness i.e. n(A)=71Let n(B) be the no. of side effects i.e.n(B) =60Let n(C) be the no. of responses to cost i.e. n(C)= 6533 responded positively to both effectiveness and side effectsSo, n(A∩B)=3331 to effectiveness and costn(A∩C)=31 28 to side effects and costn(B∩C)=2821 to none of the itemsSo, n(A∪B∪C)=140-21 = 119we 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