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Math Help - Set theory question

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    Set theory question

    5. A penny, nickel and dime are tossed together 100 times. They turn up heads 59, 52, and 47 times, respectively. The penny and nickel were heads together 33 times, the penny and dime were heads together 21 times. What can be said about the number of times that all three coins were heads together?
    I can not derive an useful conclusion from this information....
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  2. #2
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    Re: Set theory question

    Quote Originally Posted by ehpoc View Post
    I can not derive an useful conclusion from this information....
    Get a conclusion from the inclusion (exclusion principle)
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  3. #3
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    Re: Set theory question

    Hello, ehpoc!

    This is the Venn Diagram from Hell (coming soon to a theater near you).


    5. A penny, nickel and dime are tossed together 100 times.
    They turn up heads 59, 52, and 47 times, respectively.
    The penny and nickel were heads together 33 times,
    the penny and dime were heads together 21 times.
    What can be said about the number of times that all three coins were heads together?

    A Venn diagram is suggested.

    . . \begin{Bmatrix}n(P) &=& 59 \\ n(N) &=& 52 \\ n(D) &=& 37 \\ n(P\cap N) &=& 33 \\ n(P\cap D) &=& 21 \end{Bmatrix}

    Let x \,=\,n(P \cap N \cap D). .Let a \,=\,n(\overline{P} \cap N \cap D)

    From the given information, we can fill in the Venn diagram:
    Code:
                  * - - - - - - - - - - - *
                  |                       |
                  |   P                   |
                  |                       |
                  |       * - - - - - - - + - - - *
                  |  x+5  |               |       |
                  |       |               |       |
                  |       |         33-x  |       | 
          * - - - * - - - + - - - +       |       |
          |       |       |       |       |       |
          |       | 21-x  |   x   |       |       |
          |       |       |       |       | 19-a  |
          |       * - - - + - - - + - - - *       |
          |               |       |               |
          |               |   a   |           N   |
          |  26-a         |       |               |
          |               * - - - + - - - - - - - *
          |                       |
          |   D                   |
          |                       |
          * - - - - - - - - - - - *
    There is a total of 100 coins.

    . . (x+5) + (33-x) + (21-x) + x + (19-a) + (26-a) + a \;=\;100

    Hence: . 104 - a \:=\:100 \quad\Rightarrow\quad a \:=\:4


    From the center region, x, and the three adjacent regions, (33-x),\:(21-x),\:4
    . . we have: . \begin{Bmatrix}x\:\le\:33-x & [1] \\ x \:\le\:21-x & [2] \\ x\:\ge\:4 & [3]\end{Bmatrix}

    From [2]: . x \:\le\:21-x \equa\Rightarrow\quad x \:\le 10


    Therefore: . 4 \:\le\:x\:\le\:10

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