Question 721538
<pre>
{{{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,k)
locate(-.5,-2.7,C), locate(-.5,-3,PENCIL),
locate(-.3,-1,j),
locate(1.1,.4,i), 
circle(sqrt(2),sqrt(2),2), 
locate(-3.5,2.5,A),  locate(-3.9,3.5,BLACK),locate(-3.9,3,PEN),
circle(-sqrt(2),sqrt(2),2),
locate(3.5,2.5,B), locate(3,3.5,BLUE),locate(3,3,PEN),
locate(-1.3,.5,g),
locate(0,2.5,e),
locate(2,2,h),
locate(-.2,1.1,f) )}}}

There are 3 overlapping sets A,B, and C inside the big rectangle,
making 8 regions to consider.   

Circle A contains all the backpacks that had at least a black pen.
Circle B contains all the backpacks that had at least a blue pen.
Circle C contains all the backpacks that had at least a pencil.

We want to know how many are in
region k, which is the region outside all three circles.

Let's reverse the order the clues
are listed in:
</pre>
10 contained all three items.
<pre>
So we put 10 in region 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,k)
locate(-.5,-2.7,C), locate(-.5,-3,PENCIL),
locate(-.3,-1,j),
locate(1.1,.4,i), 
circle(sqrt(2),sqrt(2),2), 
locate(-3.5,2.5,A),  locate(-3.9,3.5,BLACK),locate(-3.9,3,PEN),
circle(-sqrt(2),sqrt(2),2),
locate(3.5,2.5,B), locate(3,3.5,BLUE),locate(3,3,PEN),
locate(-1.3,.5,g),
locate(0,2.5,e),
locate(2,2,h),
locate(-.2,1.1,10) )}}}
</pre>
18 contained both a blue pen and a pencil,
<pre>
We have accounted for 10 of these among the 18, so the remaining
8 are in region i.  So we put 8 in i 

{{{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,k)
locate(-.5,-2.7,C), locate(-.5,-3,PENCIL),
locate(-.3,-1,j),
locate(1.1,.4,8), 
circle(sqrt(2),sqrt(2),2), 
locate(-3.5,2.5,A),  locate(-3.9,3.5,BLACK),locate(-3.9,3,PEN),
circle(-sqrt(2),sqrt(2),2),
locate(3.5,2.5,B), locate(3,3.5,BLUE),locate(3,3,PEN),
locate(-1.3,.5,g),
locate(0,2.5,e),
locate(2,2,h),
locate(-.2,1.1,10) )}}}
</pre>
12 contained both a black pen and a pencil,
<pre>
We have accounted for 10 of these among the 12, so the remaining
2 are in region g.  So we put 2 in g

{{{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,k)
locate(-.5,-2.7,C), locate(-.5,-3,PENCIL),
locate(-.3,-1,j),
locate(1.1,.4,8), 
circle(sqrt(2),sqrt(2),2), 
locate(-3.5,2.5,A),  locate(-3.9,3.5,BLACK),locate(-3.9,3,PEN),
circle(-sqrt(2),sqrt(2),2),
locate(3.5,2.5,B), locate(3,3.5,BLUE),locate(3,3,PEN),
locate(-1.3,.5,2),
locate(0,2.5,e),
locate(2,2,h),
locate(-.2,1.1,10) )}}}
</pre>
15 contained both a black pen and a blue pen,
<pre>
We have accounted for 10 of these among the 15, so the remaining
5 are in region e.  So we put 5 in region e:

{{{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,k)
locate(-.5,-2.7,C), locate(-.5,-3,PENCIL),
locate(-.3,-1,j),
locate(1.1,.4,8), 
circle(sqrt(2),sqrt(2),2), 
locate(-3.5,2.5,A),  locate(-3.9,3.5,BLACK),locate(-3.9,3,PEN),
circle(-sqrt(2),sqrt(2),2),
locate(3.5,2.5,B), locate(3,3.5,BLUE),locate(3,3,PEN),
locate(-1.3,.5,2),
locate(0,2.5,5),
locate(2,2,h),
locate(-.2,1.1,10) )}}}
</pre>
21 contained a pencil,
<pre>
We have filled in 3 of the regions of the bottom circle 
representing all the backpacks which had at least a pencil.
2+10+8=20, so the remaining 1 of the 21 is in region j.
So we put 1 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,k)
locate(-.5,-2.7,C), locate(-.5,-3,PENCIL),
locate(-.3,-1,1),
locate(1.1,.4,8), 
circle(sqrt(2),sqrt(2),2), 
locate(-3.5,2.5,A),  locate(-3.9,3.5,BLACK),locate(-3.9,3,PEN),
circle(-sqrt(2),sqrt(2),2),
locate(3.5,2.5,B), locate(3,3.5,BLUE),locate(3,3,PEN),
locate(-1.3,.5,2),
locate(0,2.5,5),
locate(2,2,h),
locate(-.2,1.1,10) )}}}
</pre>
27 contained a blue pen
</pre>
We have filled in 3 of the regions of the right circle 
representing all the backpacks which had at least a blue pen.
5+10+8=23, so the remaining 4 of the 27 is in region h. So
we put 4 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,k)
locate(-.5,-2.7,C), locate(-.5,-3,PENCIL),
locate(-.3,-1,1),
locate(1.1,.4,8), 
circle(sqrt(2),sqrt(2),2), 
locate(-3.5,2.5,A),  locate(-3.9,3.5,BLACK),locate(-3.9,3,PEN),
circle(-sqrt(2),sqrt(2),2),
locate(3.5,2.5,B), locate(3,3.5,BLUE),locate(3,3,PEN),
locate(-1.3,.5,2),
locate(0,2.5,5),
locate(2,2,4),
locate(-.2,1.1,10) )}}}
</pre>
23 contained a black pen,
<pre>
We have filled in 3 of the regions of the left circle 
representing all the backpacks which had at least a black pen.
5+10+2=17, so the remaining 6 of the 23 is in region h.  So
we put 6 in region d.

{{{drawing(300,300,-4,4,-5,4,
rectangle(-4,-3.5,4,4),
circle(0,-.5,2),
locate(-2,2,6),
locate(-3.5,-2,k)
locate(-.5,-2.7,C), locate(-.5,-3,PENCIL),
locate(-.3,-1,1),
locate(1.1,.4,8), 
circle(sqrt(2),sqrt(2),2), 
locate(-3.5,2.5,A),  locate(-3.9,3.5,BLACK),locate(-3.9,3,PEN),
circle(-sqrt(2),sqrt(2),2),
locate(3.5,2.5,B), locate(3,3.5,BLUE),locate(3,3,PEN),
locate(-1.3,.5,2),
locate(0,2.5,5),
locate(2,2,4),
locate(-.2,1.1,10) )}}}

Now we come to the question:
</pre>
How many backpacks contained none of the three writing instruments? 
<pre>
Now we have the number in every region but k.  We are told
there are 38 backpacks.  The other 7 regions contain

10+8+2+5+1+4+6 = 36

So the remining 2 are in the outside region k.  So we
put 2 in region k:

{{{drawing(300,300,-4,4,-5,4,
rectangle(-4,-3.5,4,4),
circle(0,-.5,2),
locate(-2,2,6),
locate(-3.5,-2,2)
locate(-.5,-2.7,C), locate(-.5,-3,PENCIL),
locate(-.3,-1,1),
locate(1.1,.4,8), 
circle(sqrt(2),sqrt(2),2), 
locate(-3.5,2.5,A),  locate(-3.9,3.5,BLACK),locate(-3.9,3,PEN),
circle(-sqrt(2),sqrt(2),2),
locate(3.5,2.5,B), locate(3,3.5,BLUE),locate(3,3,PEN),
locate(-1.3,.5,2),
locate(0,2.5,5),
locate(2,2,4),
locate(-.2,1.1,10) )}}}

Answer: 2

Edwin</pre>