Question 473420
<pre><font size = 4 color = "indigo"><b>
 
Let R = the set of all gardners who grew roses, regardless
of whether they grew the other two flowers.
Let T = the set of all gardners who grew tulips, regardless
of whether they grew the other two flowers.
Let O = the set of all gardners who grew orchids, regardless
of whether they grew the other two flowers.
 
{{{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,h)
locate(0,-2.7,O),
locate(-.3,-1,g),
locate(1.1,.4,f), 
circle(sqrt(2),sqrt(2),2), 
locate(-3.5,2.5,R),
circle(-sqrt(2),sqrt(2),2),
locate(3.5,2.5,T),
locate(-1.3,.5,d),
locate(0,2.5,b),
locate(2,2,c),
locate(-.2,1.1,e) )}}}
 
The most inclusive clue we have is this one:
</pre></font></b>
4 grew all three types
<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 e, so we put 4 for "e"
 
{{{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,h)
locate(0,-2.7,O),
locate(-.3,-1,g),
locate(1.1,.4,f), 
circle(sqrt(2),sqrt(2),2), 
locate(-3.5,2.5,R),
circle(-sqrt(2),sqrt(2),2),
locate(3.5,2.5,T),
locate(-1.3,.5,d),
locate(0,2.5,b),
locate(2,2,c),
locate(-.2,1.1,red(4)) )}}}
 
Now we look at this:
</pre></font></b>
7 grew both roses and tulips
<pre><font size = 4 color = "indigo"><b>
4 of these 7 are in the middle region,
so the other 7-4 or 3 are in "b", so
we put 3 in region "b":
 
{{{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,h)
locate(0,-2.7,O),
locate(-.3,-1,g),
locate(1.1,.4,f), 
circle(sqrt(2),sqrt(2),2), 
locate(-3.5,2.5,R),
circle(-sqrt(2),sqrt(2),2),
locate(3.5,2.5,T),
locate(-1.3,.5,d),
locate(0,2.5,red(3)),
locate(2,2,c),
locate(-.2,1.1,red(4)) )}}}
 
Next we look at this clue:
</pre></font></b>
5 grew both orchids and tulips
<pre><font size = 4 color = "indigo"><b>
4 of these 5 are in the middle region,
so the other 5-4 or 1 is in "f", so
we put 1 in region "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,h)
locate(0,-2.7,O),
locate(-.3,-1,g),
locate(1.1,.4,red(1)), 
circle(sqrt(2),sqrt(2),2), 
locate(-3.5,2.5,R),
circle(-sqrt(2),sqrt(2),2),
locate(3.5,2.5,T),
locate(-1.3,.5,d),
locate(0,2.5,red(3)),
locate(2,2,c),
locate(-.2,1.1,red(4)) )}}}

Next we look at this clue:
</pre></font></b>
10 grew both roses and orchids
<pre><font size = 4 color = "indigo"><b>
4 of these 10 are in the middle region,
so the other 10-4 or 6 are in "d", 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,a),
locate(-3.5,-2,h)
locate(0,-2.7,O),
locate(-.3,-1,g),
locate(1.1,.4,red(1)), 
circle(sqrt(2),sqrt(2),2), 
locate(-3.5,2.5,R),
circle(-sqrt(2),sqrt(2),2),
locate(3.5,2.5,T),
locate(-1.3,.5,red(6)),
locate(0,2.5,red(3)),
locate(2,2,c),
locate(-.2,1.1,red(4)) )}}}

Next we look at this clue:
</pre></font></b>
45 grew roses
<pre><font size = 4 color = "indigo"><b>
3 of the regions of circle R contain 3,4,
and 6 gardenerss, that's 3+4+6=13 already accounted for,
so the remaining region "a" of circle R must
contain 45-13 = 32.  So we put 32 in region "a": 

{{{drawing(300,300,-4,4,-5,4,
rectangle(-4,-3.5,4,4),
circle(0,-.5,2),
locate(-2,2,red(32)),
locate(-3.5,-2,h)
locate(0,-2.7,O),
locate(-.3,-1,g),
locate(1.1,.4,red(1)), 
circle(sqrt(2),sqrt(2),2), 
locate(-3.5,2.5,R),
circle(-sqrt(2),sqrt(2),2),
locate(3.5,2.5,T),
locate(-1.3,.5,red(6)),
locate(0,2.5,red(3)),
locate(2,2,c),
locate(-.2,1.1,red(4)) )}}}

Next we look at this clue:
</pre></font></b>
16 grew tulips
<pre><font size = 4 color = "indigo"><b>
3 of the regions of circle T contain 3,4,
and 1 gardeners, that's 3+4+1=8 already accounted for,
so the remaining region "c" of circle T must
contain 16-8 = 8.  So we put 8 in region "c": 

{{{drawing(300,300,-4,4,-5,4,
rectangle(-4,-3.5,4,4),
circle(0,-.5,2),
locate(-2,2,red(32)),
locate(-3.5,-2,h)
locate(0,-2.7,O),
locate(-.3,-1,g),
locate(1.1,.4,red(1)), 
circle(sqrt(2),sqrt(2),2), 
locate(-3.5,2.5,R),
circle(-sqrt(2),sqrt(2),2),
locate(3.5,2.5,T),
locate(-1.3,.5,red(6)),
locate(0,2.5,red(3)),
locate(2,2,red(8)),
locate(-.2,1.1,red(4)) )}}}
 
Next we look at this clue:
</pre></font></b>
13 grew orchids
<pre><font size = 4 color = "indigo"><b>
3 of the regions of circle O contain 6,4,
and 1 gardeners, that's 6+4+1=11 already accounted for,
so the remaining region "g" of circle O must
contain 13-11 = 2.  So we put 2 in region "g": 

{{{drawing(300,300,-4,4,-5,4,
rectangle(-4,-3.5,4,4),
circle(0,-.5,2),
locate(-2,2,red(32)),
locate(-3.5,-2,h)
locate(0,-2.7,O),
locate(-.3,-1,red(2)),
locate(1.1,.4,red(1)), 
circle(sqrt(2),sqrt(2),2), 
locate(-3.5,2.5,R),
circle(-sqrt(2),sqrt(2),2),
locate(3.5,2.5,T),
locate(-1.3,.5,red(6)),
locate(0,2.5,red(3)),
locate(2,2,red(8)),
locate(-.2,1.1,red(4)) )}}}

Next we look at this clue:
</pre></font></b>
9 grew none of these three
<pre><font size = 4 color = "indigo"><b>
That is the region that is outside all three
circles, which is region "h".  So we put 9 in
region "h": 


{{{drawing(300,300,-4,4,-5,4,
rectangle(-4,-3.5,4,4),
circle(0,-.5,2),
locate(-2,2,red(32)),
locate(-3.5,-2,red(9))
locate(0,-2.7,O),
locate(-.3,-1,red(2)),
locate(1.1,.4,red(1)), 
circle(sqrt(2),sqrt(2),2), 
locate(-3.5,2.5,R),
circle(-sqrt(2),sqrt(2),2),
locate(3.5,2.5,T),
locate(-1.3,.5,red(6)),
locate(0,2.5,red(3)),
locate(2,2,red(8)),
locate(-.2,1.1,red(4)) )}}}

Now we have the numbers of gardners in all 
8 regions of the Venn diagram identified.  So
we can now answer any questions about it:
</pre>
(a) How many grew only roses? 
<pre>
These are the 32 gardners in the upper left region of R,
because those 32 are not in either of the other two circles.
Answer: 32
</pre>
(b) How many grew both roses and tulips, but not orchids? 
<pre>
There are 3+4 or 7 gardeners that are in the two parts
where circle R and circle T overlap.
Answer: 7 
</pre>
(c) How many grew only tulips? 
<pre>
These are the 8 gardners in the upper right region of T,
because those 8 are not in any of the other two circles.
Answer: 8
</pre>
(d) How many grew none of these three or only orchids?
<pre>
There are 9 who grew none of these three flowers and 
there are 2 gardners in the bottom region of O,
because those 2 are not in either of the other two 
circles.
Answer: 9+2 = 11
</pre>
(e) How many flower gardeners were surveyed?
<pre>
We add up all the gardeners in all eight regions:
32+3+8+6+4+1+2+9 = 65 
Answer: 65

Edwin</pre>