# Set theory question

• Dec 16th 2011, 08:29 AM
ehpoc
Set theory question
Quote:

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....
• Dec 16th 2011, 09:50 AM
Also sprach Zarathustra
Re: Set theory question
Quote:

Originally Posted by ehpoc
I can not derive an useful conclusion from this information....

Get a conclusion from the inclusion (exclusion principle)
• Dec 17th 2011, 09:10 AM
Soroban
Re: Set theory question
Hello, ehpoc!

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

Quote:

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.

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

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

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.

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

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

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

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

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