GATE EC 2016 Set 2 — Question 6

The question asks for the number of students who like to read books OR play sports. This means we need to find the total number of students in the union of the 'Read books' set and the 'Play sports' set, which includes students who like only reading books, only playing sports, or both.From the Venn diagram:Students who like 'Read books' only: 13
Students who like 'Read books' and 'Watch TV' only: 12
Students who like 'Read books' and 'Play sports' only: 44
Students who like 'Read books', 'Watch TV', and 'Play sports': 7Students who like 'Play sports' only: 15
Students who like 'Play sports' and 'Watch TV' only: 17To find the number of students who like to read books OR play sports, we sum all unique numbers within the 'Read books' circle and 'Play sports' circle:Number of students = (Read books only) + (Read books and Watch TV only) + (Read books and Play sports only) + (Read books, Watch TV, and Play sports) + (Play sports only) + (Play sports and Watch TV only)Number of students = $13 + 12 + 44 + 7 + 15 + 17$
Number of students = $25 + 44 + 7 + 15 + 17$
Number of students = $69 + 7 + 15 + 17$
Number of students = $76 + 15 + 17$
Number of students = $91 + 17$
Number of students = $108$Alternatively, using the Principle of Inclusion-Exclusion:
$|A \cup B| = |A| + |B| - |A \cap B|$
Where A = 'Read books' set and B = 'Play sports' set.$|\text{Read books}| = 13 + 12 + 7 + 44 = 76$
$|\text{Play sports}| = 15 + 17 + 7 + 44 = 83$
$|\text{Read books} \cap \text{Play sports}| = 44 + 7 = 51$$|\text{Read books} \cup \text{Play sports}| = 76 + 83 - 51 = 159 - 51 = 108$Therefore, the number of students who like to read books or play sports is 108.