Finding the Number of Sets in Two Groups: A Simplified Problem

[ssd]

Came to know about the following problem from a friend which can be simplified to the following:
A₁, A₂, ...A_m and B₁, B₂,...B_n are two groups of sets each group spanning the sample space.
Now there are p elements in each of A_i and each element is in exactly p₁ of the sets of the A group.
Again there are q elements in each of B_i and each element is in exactly q₁ of the sets of the B group.
We have to write, 'n' in terms of m,p,p₁,q,q₁. Thanks for any ideas.

 

[micromass]

Denote the sample space by $S$. Let us take the set $\Omega = \{(x,k)~\vert~x\in A_k\}$. Let's count the elements in $\Omega$ in two ways. In the first way, we first choose an element $x\in S$, this can be done in $|S|$ ways. Then I choose $k$ such that $x\in A_k$. This can be done in $p_1$ ways. Furthermore every choice I make yields different elements of $\Omega$. So $|\Omega| = |S| p_1$.
I can also choose $k$ first, this can be done in $m$ ways. Then I can choose $x$ such that $x\in A_k$, this can be done in $p$ ways. Every choice I made yields different elements of $\Omega$. Thus $|\Omega| = pm$. So we get $|S| p_1 = pm$. I leave the rest of the solution to you.

 

[ssd]

Thanks a lot.