# Inclusion–exclusion principle problem

172 business executives were surveyed to determine if they regularly read Fortune, Time, or Money magazines. 80 read Fortune, 70 read Time, 47 read Money, 47 read exactly two of the three magazines, 26 read Fortune and Time, 28 read Time and Money, and 7 read all three magazines. How many read none of the three magazines?

So what i tried to do was to denote Time as $$T$$, Fortune as $$F$$, and Money as $$M$$. Then i wanted to do $$172 - |T \cup F \cup M|$$. To find that amount, i tried to do the inclusion exclusion principle. I found $$|F \cup M|$$ to be $$14$$ by looking at a venn diagram ($$26-7 + 28-7 + x = 47 -> x = 7$$, then $$7+7 = 14$$), but i guess this i where things went wrong. I got $$|T \cup F \cup M| = 80 + 70 + 47 - 26 - 28 - 14 + 7$$ thus the answer would be 53, but that is false.

• Excellent reasoning! See the answer below. – S. Dolan Sep 10 '19 at 9:18

We are given explicitly that \begin{align*} |F| &= 80\\ |T| &= 70\\ |M| &= 47\\ |F\cap T| &= 26\\ |T\cap M| &= 28\\ |F\cap T\cap M| &= 7. \end{align*}

We also get that 47 read "exactly two". If you consider a Venn diagram, it is not hard to see that this corresponds to the equation \begin{align*} &|F\cap T| + |T\cap M|+ |F\cap M|-3|F\cap T\cap M|=47\\ \implies & 26 + 28 + |F\cap M| - 3(7) = 47\\ \implies & |F\cap M| = 14. \end{align*}

Thus you can plug this all in to inclusion-exclusion: \begin{align*} |F \cup T\cup M| &= |F|+|T|+|M| - |F\cap T| - |T\cap M|- |F\cap M| + |F\cap T\cap M|\\ &= 80+70+47-26-28-14+7=136, \end{align*}

and thus the answer is $$172-136=\boxed{\text{36 people}}$$.

In your expression $$|T \cup F \cup M| = 80 + 70 + 47 - 26 - 28 - 14 + 7=136$$.

This gives a solution of $$36$$ not $$53$$.

We have:

$$|T \cup F \cup M| = |T| + |F| + |M| - |T \cap F| - |T \cap M| - |F \cap M| + |T \cap F \cap M|$$ $$= 70 + 80 + 47 - 26 - 28 - 14 + 7 = 136$$

Then, the number of people who do not read any magazine equals:

$$172 - |T \cup F \cup M| = 36$$