# Need help on simple problem

#### steelmaste

can someone show me the combinatorics steps to solve this problem ? I can solve it really easily by drawing a graph,
but my teacher said she would not allow us drawing and wants us to play with the combinatrics of probability to arrive at the answer, I've been trying to solve it a few times already but still cant

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#### TKHunny

Have you considered drawing the three intersecting circles and filling in the 7 individual spaces?

#### steelmaste

that's the easy way to solve it and I did that ,but my teacher said that it is not allowed to do that on my exam so I have to do it the long way. But when I try to do it the long way I always fail... So now I'm trying to seek help.

#### Plato

MHF Helper
that's the easy way to solve it and I did that ,but my teacher said that it is not allowed to do that on my exam so I have to do it the long way. But when I try to do it the long way I always fail... So now I'm trying to seek help.
Are you telling us that your instructor is telling a class that using Venn diagrams is not the best way to do this?
We have eight distinct sets:
$\mathscr{U}$ is the universal set and $\|\mathscr{U}\|=10^5$ the total.
$G=I\cap II\cap III$ and $\|G\|=10^3$
$D=I\cap II\setminus G$ and $\|D\|=7\cdot 10^3$
$E=II\cap III\setminus G$ and $\|E\|=3\cdot 10^3$
$F=I\cap III\setminus G$ and $\|F\|=1\cdot 10^3$
$A=I\setminus(II\cup III)$ and $\|A\|=1\cdot 10^3$
$B=II\setminus(I\cup III)$ and $\|B\|=19\cdot 10^3$
$C=I\setminus(I\cup II)$ and $\|C\|=0$

The answer for a) is $\|A\cup B\cup C\|=?$.

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