```Question 288071
<pre><font size = 4 color = "indigo"><b>

Let A = the set of all students who liked algebra, regardless
of whether they liked the other two subjects.
Let G = the set of all students who liked geometry, regardless
of whether they liked the other two subjects.
Let C = the set of all students who liked calculus, regardless
of whether they liked the other two subjects.

{{{drawing(300,300,-4,4,-5,4,
rectangle(-4,-3.5,4,4),
circle(0,-.5,2),
locate(-2,2,d),
locate(-3.5,-2,m)
locate(0,-2.7,C),
locate(-.3,-1,k),
locate(1.1,.4,j),
circle(sqrt(2),sqrt(2),2),
locate(-3.5,2.5,A),
circle(-sqrt(2),sqrt(2),2),
locate(3.5,2.5,G),
locate(-1.3,.5,h),
locate(0,2.5,e),
locate(2,2,i),
locate(-.2,1.1,f) )}}}

The most inclusive clue we have is this one:
</pre></font></b>
4 liked calculus, algebra, and geometry
<pre><font size = 4 color = "indigo"><b>
Those 4 are in all three circles, and the only
region that is common to all three circles is
the middle region f, so we put 4 for f

{{{drawing(300,300,-4,4,-5,4,
rectangle(-4,-3.5,4,4),
circle(0,-.5,2),
locate(-2,2,d),
locate(-3.5,-2,m)
locate(0,-2.7,C),
locate(-.3,-1,k),
locate(1.1,.4,j),
circle(sqrt(2),sqrt(2),2),
locate(-3.5,2.5,A),
circle(-sqrt(2),sqrt(2),2),
locate(3.5,2.5,G),
locate(-1.3,.5,h),
locate(0,2.5,e),
locate(2,2,i),
locate(-.2,1.1,red(4)) )}}}

Now we look at this:
</pre></font></b>
12 liked calculus and geometry
<pre><font size = 4 color = "indigo"><b>
4 of these 12 are in the middle region,
so the other 12-4 or 8 are in j, so
we put 8 in region j:

{{{drawing(300,300,-4,4,-5,4,
rectangle(-4,-3.5,4,4),
circle(0,-.5,2),
locate(-2,2,d),
locate(-3.5,-2,m)
locate(0,-2.7,C),
locate(-.3,-1,k),
locate(1.1,.4,red(8)),
circle(sqrt(2),sqrt(2),2),
locate(-3.5,2.5,A),
circle(-sqrt(2),sqrt(2),2),
locate(3.5,2.5,G),
locate(-1.3,.5,h),
locate(0,2.5,e),
locate(2,2,i),
locate(-.2,1.1,red(4)) )}}}

Next we look at this clue:
</pre></font></b>
18 liked calculus but not algebra
<pre><font size = 4 color = "indigo"><b>
So 18 are in circle C but not in circle A.

8 of these 18 we just got through finding so
the other 18-8 or 10 must be in region k,
since they do not like algebra.  So we put
10 in region k:

{{{drawing(300,300,-4,4,-5,4,
rectangle(-4,-3.5,4,4),
circle(0,-.5,2),
locate(-2,2,d),
locate(-3.5,-2,m)
locate(0,-2.7,C),
locate(-.3,-1,red(10)),
locate(1.1,.4,red(8)),
circle(sqrt(2),sqrt(2),2),
locate(-3.5,2.5,A),
circle(-sqrt(2),sqrt(2),2),
locate(3.5,2.5,G),
locate(-1.3,.5,h),
locate(0,2.5,e),
locate(2,2,i),
locate(-.2,1.1,red(4)) )}}}

Next we look at:
</pre></font></b>
25 liked calculus
<pre><font size = 4 color = "indigo"><b>
Now we can complete the circle C,
because we have accounted for the
numbers in three of its four regions
of its regions, that's 4+8+10, or 22,
so the other 25-22 or 3 are in region h.
So we put 3 in region h:

{{{drawing(300,300,-4,4,-5,4,
rectangle(-4,-3.5,4,4),
circle(0,-.5,2),
locate(-2,2,d),
locate(-3.5,-2,m)
locate(0,-2.7,C),
locate(-.3,-1,red(10)),
locate(1.1,.4,red(8)),
circle(sqrt(2),sqrt(2),2),
locate(-3.5,2.5,A),
circle(-sqrt(2),sqrt(2),2),
locate(3.5,2.5,G),
locate(-1.3,.5,red(3)),
locate(0,2.5,e),
locate(2,2,i),
locate(-.2,1.1,red(4)) )}}}

Next we look at this clue:
</pre></font></b>
10 liked algebra but neither calculus nor geometry
<pre><font size = 4 color = "indigo"><b>
These 10 are all in region d because they are the
only ones in the algebra circle who are not also
in either of the other two circles.  So we can
put 10 in region d

{{{drawing(300,300,-4,4,-5,4,
rectangle(-4,-3.5,4,4),
circle(0,-.5,2),
locate(-2,2,red(10)),
locate(-3.5,-2,m)
locate(0,-2.7,C),
locate(-.3,-1,red(10)),
locate(1.1,.4,red(8)),
circle(sqrt(2),sqrt(2),2),
locate(-3.5,2.5,A),
circle(-sqrt(2),sqrt(2),2),
locate(3.5,2.5,G),
locate(-1.3,.5,red(3)),
locate(0,2.5,e),
locate(2,2,i),
locate(-.2,1.1,red(4)) )}}}

Next clue:
</pre></font></b>
2 liked geometry and algebra but not calculus.
<pre><font size = 4 color = "indigo"><b>
These 2 are in region e because that region is
part of the geometry and algebra circles but
not part of the calculus circle,  So we put
2 in region e:

{{{drawing(300,300,-4,4,-5,4,
rectangle(-4,-3.5,4,4),
circle(0,-.5,2),
locate(-2,2,red(10)),
locate(-3.5,-2,m)
locate(0,-2.7,C),
locate(-.3,-1,red(10)),
locate(1.1,.4,red(8)),
circle(sqrt(2),sqrt(2),2),
locate(-3.5,2.5,A),
circle(-sqrt(2),sqrt(2),2),
locate(3.5,2.5,G),
locate(-1.3,.5,red(3)),
locate(0,2.5,red(2)),
locate(2,2,i),
locate(-.2,1.1,red(4)) )}}}

Next clue:
</pre></font></b>
15 liked geometry
<pre><font size = 4 color = "indigo"><b>
So there are 15 students in the geometry cirle,
and we have accounted for 2+4+8 or 14 of the 15
in the geometry circle, so that leaves 15-14 or
just 1 student in region i. So we put 1 student
in region i:

{{{drawing(300,300,-4,4,-5,4,
rectangle(-4,-3.5,4,4),
circle(0,-.5,2),
locate(-2,2,red(10)),
locate(-3.5,-2,m)
locate(0,-2.7,C),
locate(-.3,-1,red(10)),
locate(1.1,.4,red(8)),
circle(sqrt(2),sqrt(2),2),
locate(-3.5,2.5,A),
circle(-sqrt(2),sqrt(2),2),
locate(3.5,2.5,G),
locate(-1.3,.5,red(3)),
locate(0,2.5,red(2)),
locate(2,2,red(1)),
locate(-.2,1.1,red(4)) )}}}

Finally we look at,
</pre></font></b>
In an interview of 50 math majors,
<pre><font size = 4 color = "indigo"><b>
So of these 50 math majors, we have accounted for
10+2+1+3+4+8+10=38 of them, so the remaining 50-38
or 12 math majors are in the region outside all
three circles, which is m.  So we put 12 in region
m.

{{{drawing(300,300,-4,4,-5,4,
rectangle(-4,-3.5,4,4),
circle(0,-.5,2),
locate(-2,2,red(10)),
locate(-3.5,-2,red(12)),
locate(0,-2.7,C),
locate(-.3,-1,red(10)),
locate(1.1,.4,red(8)),
circle(sqrt(2),sqrt(2),2),
locate(-3.5,2.5,A),
circle(-sqrt(2),sqrt(2),2),
locate(3.5,2.5,G),
locate(-1.3,.5,red(3)),
locate(0,2.5,red(2)),
locate(2,2,red(1)),
locate(-.2,1.1,red(4)) )}}}

So 12 math majors didn't like any of
the three math subjects, so those 12
had better change their major to history
or English or a foreign language, don't
you think!!

Anyway you only have to answer this:

Of those surveyed, how many liked calculus and algebra?
<pre><font size = 4 color = "indigo"><b>
Those are the students in the two region with 3
and 4 in them for they are in both the calculus and
the algebra circles. So the only thing you have to