Q.In a class of 200 students who appeared certain examinations, 35 students failed in CET, 40 in NEET and 40 in JEE, 20 failed in CET and NEET, 17 in NEET and JEE, 15 in CET and JEE and 5 failed in all three examinations. Find how many students did not fail in any examination.

This concept covers the CARDINALITY (counting) formulas that let you find the SIZE of a union or intersection without having to list out every element — essential for word problems involving surveys, exam results, or consumer preferences given only as totals and overlaps. For two sets: n(A∪B) = n(A) + n(B) − n(A∩B) — you add the two individual counts, then subtract the overlap once (since a direct sum of n(A)+n(B) would double-count every shared element). When A and B are disjoint, this simplifies to n(A∪B) = n(A)+n(B). Related identities let you isolate any one piece given the others, e.g. n(A∩B') = n(A) − n(A∩B) (the part of A outside B). For THREE sets, the inclusion-exclusion formula extends naturally: n(A∪B∪C) = n(A)+n(B)+n(C) − n(A∩B) − n(B∩C) − n(A∩C) + n(A∩B∩C) — each pairwise overlap, having been counted twice in the initial sum, is subtracted once; the triple overlap, having then been subtracted three times too many, is added back once. This formula (and its two-set version) is the standard tool for problems like 'how man …