Let D = the set of those who drove alone.
Let C = the set of those who rode in a carpool.
Let P = the set of those who rode public transportation.
There are 8 regions (subsets) which I have labeled with small
letters s thru z:
The most inclusive clue we have is this one:
3 used all three methods
Those 3 are in all three circles, and the only
region that is common to all three circles is
the middle region w, so we put 3 for w
Now we look at this:
6 used both carpool and public transportation.
3 of those 6 are already accounted for in the middle region,
so that leaves 6-3 or 3 to go in the other region common
to circles C and P, which is region x, so we put 3 in that region:
Now we look at this:
4 used both a carpool (C) and sometimes their own cars (D).
3 of those 4 are already accounted for in the middle region,
so that leaves 4-3 or 1 to go in the other region common
to circles C and D, which is region t, so we put 1 in that region:
Now we look at this:
30 rode in a carpool.
We have numbers in 3 of the regions of circle C, so
1+3+3 or 7 of those 30 are already accounted
for so that leaves 30-7 or 23 to go in the remaining
region of circle C, region u, so we put 23 in that region:
Now we have completed all four regions of circle C,
and see that if we add up all four regions we have
1+3+3+23 we have 30 in circle C, those who rode in
a carpool.
Now we look at this:
8 used buses (public transportation) as well as their own cars (drove).
3 of those 8 are already accounted for in the middle region,
so that leaves 8-3 or 5 to go in the other region common
to circles P and D, which is region v, so we put 5 in that region:
Now we look at this:
36 said they drove alone.
We have numbers in 3 of the regions of circle D, so
1+3+5 or 9 of those 36 are already accounted
for so that leaves 36-9 or 27 to go in the remaining
region of circle D, region s, so we put 27 in that region:
Now we have completed all four regions of circle D,
and see that if we add up all four regions we have
27+1+3+5 we have 36 in circle C, those who drove
themselves.
Now we look at this:
32 rode public transportation.
We have numbers in 3 of the regions of circle P, so
5+3+3 or 11 of those 32 are already accounted
for so that leaves 32-11 or 21 to go in the remaining
region of circle P, region y, so we put 21 in that region:
Now we know how many students are in all but one of
the 8 regions, the region labeled z, that is outside
all the circles. These are what the problem asks
for, how many are not using any of these.
To find this out we look at this:
Now we look at this:
A poll was taken of 100 students
We have accounted for 27+1+23+5+3+3+21 or 83 of the
100 students who use one or more of the three kind of
transportation mentioned. The others are outside
all the circles, in region z. So the number in
the region outside the circles, region z is
100-83 or 17.
So the answer is 17.
Edwin
