NIMCET 2009 Question Paper with solutions

Question : 35 of 120

Marks: +1, -0

A_{1}, A_{2}, A_{3}

and

A_{4}

are subsets of a set

U

containing 75 elements with the following properties : Each subset contains 28 elements; the intersection of any two of the subsets contains 12 elements; the intersection of any three of the subsets contains 5 elements; the intersection of all four subsets contains 1 element. The number of elements belonging to none of the four subsets is

Validate

Solution:

n(A_{1} ∪ A_{2} ∪ A_{3} ∪ A_{4})

=∑↙{n } (A_{i})-{∑\;∑}↙{i < j}n[A_{i} ∩ A_{j})+{∑∑ ∑}↙{i < j < k}(A_{i} ∩ A_{j} ∩ A_{k})

-{∑ ∑ ∑ ∑}↙{i < J < k < l} A_{i} ∩ A_{i} ∩ A_{k} ∩ A_{1}

=4 × 28-{ }^{4} {C}_{2} × 12+{ }^{4} C_{3} × 5-{ }^{4} C_{4} × 1

=112-72+20-1=59

The number of elements belonging to none of the four subsets

=75-59=16

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