let s be a set with N elements and A1,...,A101 be 101 (possibly not disjoint) subsets of S

Question

So the question I'm having problem with is the following: let s be a set with N elements and A1,...,A101 be 101 (possibly not disjoint) subsets of S with the following 5 properties:

1. each elements of S belongs to at least one subset of Ai ∈ {A1,...,A101},

2. each susbset Ai∈ {A1,...,A101} contains exactly 1000 elements of S,

3. the intersection of any pair {Ai,Aj}of distinct subsets of {A1,...,A101} contains exactly 200 elements,

4. the intersectin of any 3 distinct subsets {Ai,Aj,AK} of PA1,...,A101} contains exactly 6 elements

5. the intersection of any 4 or more distinct subsets of {A1,...,A101} empty

Using the inclusion/exclusion principle, compute N and cardinality of S.

I really don't know how to approach this problem.. Step by step would be very much appreciated thank you

Answer 1

Hint. By your first condition we have $$S=A_1\cup A_2\cup\cdots\cup A_{101}\ .$$ The inclusion/exclusion formula for $101$ sets is $$\eqalign{ N=|S| &=|A_1|+|A_2|+\cdots+|A_{101}|\cr &\qquad {}-|A_1\cap A_2|-\cdots\cr &\qquad {}+|A_1\cap A_2\cap A_3|+\cdots\cr &\qquad {}-|A_1\cap A_2\cap A_3\cap A_4|+\cdots\cr &\qquad {}+|A_1\cap A_2\cap A_3\cap A_4\cap A_5|+\cdots\cr &\qquad {}-\cdots\ .\cr}$$ Now the first line on the right hand side contains $101$ terms, each equal to $1000$, for a total of $101000$. See if you can do a similar calculation

• for the second line;
• for the third line;
• and then see what happens for the fourth and all subsequent lines.

Good luck!