Question Detail
Question
In a survey of 290 newspapers readers, 181 of them read the Daily Times, 142 read The Guardian, 117 read The Punch and each reads at least one of the three papers. If 75 read the Daily Times and The Guardian and 60 read the Daily Times and The Punch and 54 read The Guardian and The Punch. How many readers read exactly two of the papers?
Correct Answer
Option B is the correct answer.
Detailed Explanation
Let D, G, P represent Daily Times, Guardian, Punch. n(D)=181, n(G)=142, n(P)=117, n(D∩G)=75, n(D∩P)=60, n(G∩P)=54, n(D∪G∪P)=290 (since each reads at least one). Number who read exactly two = n(D∩G) + n(D∩P) + n(G∩P) - 3n(D∩G∩P). First, find n(D∩G∩P) using inclusion-exclusion: n(D∪G∪P) = n(D)+n(G)+n(P) - [n(D∩G)+n(D∩P)+n(G∩P)] + n(D∩G∩P). So 290 = 181+142+117 - (75+60+54) + x = 440 - 189 + x = 251 + x => x = 39. Then exactly two = (75+60+54) - 3(39) = 189 - 117 = 72.
Hint
Use inclusion-exclusion for three sets. Exactly two = sum of pairwise intersections minus 3 times the triple intersection.