Question 1185492
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A poll was taken of 100 students at a commuter campus to find out how they got to campus. The results were as follows:
28 said they drove alone.
31 rode in a carpool.
30 rode public transportation.
7 used both carpools and public transportation.
3 used both a carpool and sometimes their own cars.
5 used buses as well as their own cars.
2 used all three methods.
How many used none of the above-mentioned means of transportation?
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<pre>
This problem is on the Inclusion-Exclusion pronciple.


You are given the universal set U of 100 students an 3 basic subsets inside it:


    subset A of 28 elements ("alone")

    subset C of 31 elements ("carpool")

    subset P of 30 elements ("public transportation")


You also given their in-pair intersections  


    subset AC of 3 elements (used both a carpool and sometimes their own cars)

    subset CP of 7 elements (used both carpools and public transportation)
    
    subset AP of 5 elements (used buses as well as their own cars)


Finally, you are given that the triple intersectio ACP  has 2 elements.


Using the exclusive-inclusive priinciple, you can find the number of students in the UNION of the subsets A U C U P


    n(A U C U P) = n(A) + n(c) + n(P) - n(AC) - n(AP) - n(CP) + n(ACP) = 

                 =  28  + 31   +30    -   3   -  7    -  5    +  2 = 76.


The rest,   100 - 76 = 24  belong to the cathegory under the problem's question.
</pre>

Solved.


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On inclusion-exclusion principle, &nbsp;see this Wikipedia article


https://en.wikipedia.org/wiki/Inclusion%E2%80%93exclusion_principle



To see many other similar &nbsp;(and different) &nbsp;solved problems, &nbsp;see the 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>

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

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

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF =https://www.algebra.com/algebra/homework/Permutations/Inclusion-Exclusion-principle.lesson>Inclusion-Exclusion principle problems</A> 


in this site.



Happy learning (!)