Question 616847
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 them How many students are in the class?

<pre><font size = 4 color = "indigo"><b>

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.]

{{{drawing(300,300,-4,4,-5,4,
rectangle(-4,-3.5,4,4),
circle(0,-.5,2),
locate(-2,2,a),
locate(-3.5,-2,i)
locate(0,-2.7,B),
locate(-.3,-1,h),
locate(1.1,.4,g),
circle(sqrt(2),sqrt(2),2),
locate(-3.5,2.5,P),
circle(-sqrt(2),sqrt(2),2),
locate(3.5,2.5,T),
locate(-1.3,.5,e),
locate(0,2.5,c),
locate(2,2,d),
locate(-.2,1.1,f) )}}}

The most inclusive clue we have is this one:
</pre></font></b>
>>...3 on all three,...<<
<pre><font size = 4 color = "indigo"><b>
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

{{{drawing(300,300,-4,4,-5,4,
rectangle(-4,-3.5,4,4),
circle(0,-.5,2),
locate(-2,2,a),
locate(-3.5,-2,i)
locate(0,-2.7,B),
locate(-.3,-1,h),
locate(1.1,.4,g),
circle(sqrt(2),sqrt(2),2),
locate(-3.5,2.5,P),
circle(-sqrt(2),sqrt(2),2),
locate(3.5,2.5,T),
locate(-1.3,.5,e),
locate(0,2.5,c),
locate(2,2,d),
locate(-.2,1.1,3) )}}}

The most EXcluding clue we have is this one:
</pre></font></b>
>>...and 1 on none of them...<<
<pre><font size = 4 color = "indigo"><b>
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:

{{{drawing(300,300,-4,4,-5,4,
rectangle(-4,-3.5,4,4),
circle(0,-.5,2),
locate(-2,2,a),
locate(-3.5,-2,1)
locate(0,-2.7,B),
locate(-.3,-1,h),
locate(1.1,.4,g),
circle(sqrt(2),sqrt(2),2),
locate(-3.5,2.5,P),
circle(-sqrt(2),sqrt(2),2),
locate(3.5,2.5,T),
locate(-1.3,.5,e),
locate(0,2.5,c),
locate(2,2,d),
locate(-.2,1.1,3) )}}}

Now we look at this:
</pre></font></b>
>>...9 on a plane and train,...<<
<pre><font size = 4 color = "indigo"><b>
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:

{{{drawing(300,300,-4,4,-5,4,
rectangle(-4,-3.5,4,4),
circle(0,-.5,2),
locate(-2,2,a),
locate(-3.5,-2,1)
locate(0,-2.7,B),
locate(-.3,-1,h),
locate(1.1,.4,g),
circle(sqrt(2),sqrt(2),2),
locate(-3.5,2.5,P),
circle(-sqrt(2),sqrt(2),2),
locate(3.5,2.5,T),
locate(-1.3,.5,e),
locate(0,2.5,6),
locate(2,2,d),
locate(-.2,1.1,3) )}}}

Now we look at this:
</pre></font></b>
>>...12 on a train and boat,...<<
<pre><font size = 4 color = "indigo"><b>
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:

{{{drawing(300,300,-4,4,-5,4,
rectangle(-4,-3.5,4,4),
circle(0,-.5,2),
locate(-2,2,a),
locate(-3.5,-2,1)
locate(0,-2.7,B),
locate(-.3,-1,h),
locate(1.1,.4,9),
circle(sqrt(2),sqrt(2),2),
locate(-3.5,2.5,P),
circle(-sqrt(2),sqrt(2),2),
locate(3.5,2.5,T),
locate(-1.3,.5,e),
locate(0,2.5,6),
locate(2,2,d),
locate(-.2,1.1,3) )}}}

Now we look at this:
</pre></font></b>
>>...14 on a plane and boat,...<<
<pre><font size = 4 color = "indigo"><b>

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:
{{{drawing(300,300,-4,4,-5,4,
rectangle(-4,-3.5,4,4),
circle(0,-.5,2),
locate(-2,2,a),
locate(-3.5,-2,1)
locate(0,-2.7,B),
locate(-.3,-1,h),
locate(1.1,.4,9),
circle(sqrt(2),sqrt(2),2),
locate(-3.5,2.5,P),
circle(-sqrt(2),sqrt(2),2),
locate(3.5,2.5,T),
locate(-1.3,.5,11),
locate(0,2.5,6),
locate(2,2,d),
locate(-.2,1.1,3) )}}}

Now we look at this:
</pre></font></b>
>>...25 students have been on a plane,...<<
<pre><font size = 4 color = "indigo"><b>

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:

{{{drawing(300,300,-4,4,-5,4,
rectangle(-4,-3.5,4,4),
circle(0,-.5,2),
locate(-2,2,5),
locate(-3.5,-2,1)
locate(0,-2.7,B),
locate(-.3,-1,h),
locate(1.1,.4,9),
circle(sqrt(2),sqrt(2),2),
locate(-3.5,2.5,P),
circle(-sqrt(2),sqrt(2),2),
locate(3.5,2.5,T),
locate(-1.3,.5,11),
locate(0,2.5,6),
locate(2,2,d),
locate(-.2,1.1,3) )}}}

Now we look at this:
</pre></font></b>
>>...20 on a train,...<<
<pre><font size = 4 color = "indigo"><b>
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:
{{{drawing(300,300,-4,4,-5,4,
rectangle(-4,-3.5,4,4),
circle(0,-.5,2),
locate(-2,2,5),
locate(-3.5,-2,1)
locate(0,-2.7,B),
locate(-.3,-1,h),
locate(1.1,.4,9),
circle(sqrt(2),sqrt(2),2),
locate(-3.5,2.5,P),
circle(-sqrt(2),sqrt(2),2),
locate(3.5,2.5,T),
locate(-1.3,.5,11),
locate(0,2.5,6),
locate(2,2,2),
locate(-.2,1.1,3) )}}}

Now we look at this:
</pre></font></b>
>>...27 on a boat,...<<
<pre><font size = 4 color = "indigo"><b>
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:
{{{drawing(300,300,-4,4,-5,4,
rectangle(-4,-3.5,4,4),
circle(0,-.5,2),
locate(-2,2,5),
locate(-3.5,-2,1)
locate(0,-2.7,B),
locate(-.3,-1,4),
locate(1.1,.4,9),
circle(sqrt(2),sqrt(2),2),
locate(-3.5,2.5,P),
circle(-sqrt(2),sqrt(2),2),
locate(3.5,2.5,T),
locate(-1.3,.5,11),
locate(0,2.5,6),
locate(2,2,2),
locate(-.2,1.1,3) )}}}

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:
</pre></font></b>
>>...How many students are in the class?...<<
<pre><font size = 4 color = "indigo"><b>
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</pre>