Question 1091470
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According to a survey, 16 people liked ham, 19 liked eggs, 18 liked mackerel, 
8 liked mackerel and eggs, 
5 liked ham and eggs, 
7 liked ham and mackerel, 
and 3 liked all three foods. 


<pre>
1)  how many liked only mackerel?                     18 - 8 - 7 + 3.


2)  how many liked at least two of three foods?        8 + 5 + 7 - 2*3.


3)  how many liked only one of three foods?           16 + 19 + 18 - 8 - 5 - 7 + 2*3.


4)  how many liked eggs and mackerel but not ham?     19 + 18 - 8 + 3.


5)  how many people were surveyed?                    16 + 19 + 18 - 8 - 5 - 7 + 3.
</pre>


First, &nbsp;I put the numbers only in hope that they will explain everything to you without my words.



But then I put some minimal explanations to worm your mind.


1) &nbsp;how many liked only mackerel ?                     

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Take those who like mackerel; &nbsp;subtract those who like ME; &nbsp;subtract those who like MH; &nbsp;add MHE what were subtracted twice.



2) &nbsp;how many liked at least two of three foods ?        
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;ME + HE + HM - 2*HEM = 8 + 5 + 7 - 2*3. 


&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(when we add ME + HE + HM, we count the triple intersection thrice; therefore, I subtract HEM twice . . . )



The same or similar logic works in other cases . . . 



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To get familiar with the subject, &nbsp;look into my lessons

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/word/misc/Counting-elements-in-sub-sets-of-a-given-finite-set.lesson>Counting elements in sub-sets of a given finite set</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/word/misc/Advanced-probs-counting-elements-in-sub-sets-of-a-given-finite-set.lesson>Advanced problems on counting elements in sub-sets of a given finite set</A>

in this site.