SOLUTION: In a class, 25 students have been on a plane, 20 on a train, 27 on a boat, 9 on a plane and train, 12 on a train and boat, 14 on a plane and boat, 3 on all three, and 1 on none of

Let P = the set of all students who have been on a plane, regardless
of whether they have been on a train or a boat.
Let T = the set of all students who have been on a train, regardless
of whether they have been on a plane or a boat.
Let B = the set of all students who have been on a boat, regardless
of whether they have been on a plane or a train.


There are 8 regions which I have labeled with arbitrary small
letters. [I skipped little "b" because I used capital "B" for the
set of all students who have been on a boat.]



The most inclusive clue we have is this one:

Those 3 are in all three circles, and the only
region that is common to all three circles is
the middle region f, so we put 3 for f



The most EXcluding clue we have is this one:

That 1 is outside all three circles, and the only
region outside all three circles is region i, so we 
put 1 in region i:



Now we look at this:

3 of those 9 are already accounted for in the middle region,
so that leaves 9-3 or 6 to go in the other region common
to circles P and T, which is region c, so we put 6 in that region:



Now we look at this:

3 of those 12 are already accounted for in the middle region,
so that leaves 12-3 or 9 to go in the other region common
to circles T and B, which is region g, so we put 9 in that region:



Now we look at this:

3 of those 14 are already accounted for in the middle region,
so that leaves 14-3 or 11 to go in the other region common
to circles P and B, which is region e, so we put 11 in that region:


Now we look at this:

We have numbers in 3 of the regions of circle P, so 
6+3+11 or 20 of those 25 are already accounted
for so that leaves 25-20 or 5 to go in the remaining
region of circle P, region a, so we put 5 in that region:



Now we look at this:

We have numbers in 3 of the regions of circle T, so 
6+3+9 or 18 of those 20 are already accounted
for so that leaves 20-18 or 2 to go in the remaining
region of circle T, region d, so we put 2 in that region:


Now we look at this:

We have numbers in 3 of the regions of circle B, so 
11+3+9 or 23 of those 27 are already accounted
for so that leaves 27-23 or 4 to go in the remaining
region of circle B, region h, so we put 4 in that region:


We are now ready to look at the question, for now
we can answer anything we are asked because we have
the complete Venn diagram:

Just add up all the 8 numbers in the regions:

5+6+2+11+3+9+4+1 = 41

Answer: There are 41 students in the class.

Edwin