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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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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.
Solved.
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On inclusion-exclusion principle, see this Wikipedia article
https://en.wikipedia.org/wiki/Inclusion%E2%80%93exclusion_principle
To see many other similar (and different) solved problems, see the lessons
- Counting elements in sub-sets of a given finite set
- Advanced problems on counting elements in sub-sets of a given finite set
- Challenging problems on counting elements in subsets of a given finite set
- Selected problems on counting elements in subsets of a given finite set
- Inclusion-Exclusion principle problems
in this site.
Happy learning (!)