Explanation
1. Given Data
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Total elements in Universal set n(U)=75
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Number of elements in each subset n(Ai)=28
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Number of elements in the intersection of any two subsets n(Ai∩Aj)=12
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Number of elements in the intersection of any three subsets n(Ai∩Aj∩Ak)=5
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Number of elements in the intersection of all four subsets n(A1∩A2∩A3∩A4)=1
2. Formula for the Union of Four Sets
Using the Principle of Inclusion-Exclusion for four sets:
n(A1∪A2∪A3∪A4)=∑n(Ai)−∑n(Ai∩Aj)+∑n(Ai∩Aj∩Ak)−n(A1∩A2∩A3∩A4)
3. Calculate each component
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Sum of individual sets: There are (14)=4 such sets.
∑n(Ai)=4×28=112
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Sum of intersections of two sets: There are (24)=6 such pairs.
∑n(Ai∩Aj)=6×12=72
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Sum of intersections of three sets: There are (34)=4 such triplets.
∑n(Ai∩Aj∩Ak)=4×5=20
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Intersection of all four sets: There is (44)=1 such intersection.
n(A1∩A2∩A3∩A4)=1
4. Find the total number of elements in the union
n(A1∪A2∪A3∪A4)=112−72+20−1
n(A1∪A2∪A3∪A4)=40+20−1
5. Find the elements belonging to none of the subsets
The number of elements in none of the subsets is the complement of the union:
None=n(U)−n(A1∪A2∪A3∪A4)
Correct Option:
(c) 16
