SOLUTION: 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

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.