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I Counting the number of sets.

  1. Mar 3, 2016 #1

    ssd

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    Came to know about the following problem from a friend which can be simplified to the following:
    A1, A2, ....Am and B1, B2,...Bn are two groups of sets each group spanning the sample space.
    Now there are p elements in each of Ai and each element is in exactly p1 of the sets of the A group.
    Again there are q elements in each of Bi 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.
     
  2. jcsd
  3. Mar 3, 2016 #2

    micromass

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    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.
     
  4. Mar 3, 2016 #3

    ssd

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    Thanks a lot.
     
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