A poll was taken of 100 students at a commuter campus to find out how they got to campus. The results were as follows: 31 said they drove alone, 39 rode in carpool, 35 rode public transportation, 10 used both carpool and public transportation, 7 used both a carpool and sometimes their own cars, 9 used buses as well as their own cars, 5 used all three methods. how many used none of the above-mentioned means of transportation?

Answer 1

Make a Venn diagram, showing three overlapping circles, label them
A for driving alone
C for carpool
P for public trans

put 5 in the intersection of all 3 circles
10 used C and P, so 10 goes in that intersection, BUT 5 are already counted, so place 5 in the outer part of the intersection of C and P
in the same way, place 4 in the outer part of A and P
in the same way , place 2 in the outer part of A and C
now each circle itself
A should have a total of 31, but I already have 11 counting in the A circle, leaving 20 in the unused part of A
C should have 39, but I already have 12 filled in, leaving 27 for the rest of circle C
P should have 35, but I already have 14 filled in, leaving 21 for the rest of circle P

Adding up all the numbers I see filled in, I get 84
so from the 100 students, 16 do not use any of the 3 methods.

check my arithmetic

Answer 2

N(A and C and P)
= N(A) + N(C) + N(P) - N(A and C) - N(A and P) - N(C and P) + N(A and C and P)
= 31 + 39 + 35 - 10 - 7 - 9 + 5
= 84

so 100 - 84 are unaccounted for
16 don't use any of the given ways