<PRE><B>Question 1149313</B>
&lt;br&gt;
The questions that are asked, if the problem is read exactly as written, are impossible to answer.&lt;br&gt;
The problem asks us to &quot;find those who study all 3 subjects&quot; and &quot;find (those) who study only mathematics and chemistry&quot;.&lt;br&gt;
The only information given in the problem is the NUMBERS of students that take different courses; there are no names of students given, so it is not possible to know WHICH students take various courses.&lt;br&gt;
Note further that the information given in the problem is open to different interpretations.
(1) The last number given should say that 15 students study ONLY one of the three subjects.
(2) The statement of the problem also should specifically state that there are no students who take none of the three courses, or, alternatively, that each of the 36 students takes at least one of them.&lt;br&gt;
However, the problem AS IT WAS INTENDED TO BE PRESENTED is an interesting one.&lt;br&gt;
There are 7 combinations of at least one of the three subjects Chemistry (C), Mathematics (M), and Physics (P) that a student can take:&lt;br&gt;&lt;pre&gt;
CMP
CM
CP
MP
C
M
P&lt;br&gt;&lt;/pre&gt;
Let x be the number who study all three subjects.&lt;br&gt;
Then &quot;17 study physics and mathematics&quot; means (17-x) study ONLY physics and mathematics&lt;br&gt;
And &quot;18 study physics and chemistry&quot; means (18-x) study ONLY physics and chemistry.&lt;br&gt;
Now, knowing that a total of 25 students study physics, we can find an expression for the number that study ONLY physics.
{{{25-((x)+(18-x)+(17-x))}}} which simplifies to 10-x.&lt;br&gt;
At this point, we have nowhere else to go without introducing another unknown.&lt;br&gt;
Let y be the number that study only chemistry and mathematics.  Then we can get expressions for the numbers that study only chemistry or only mathematics.
only chemistry: {{{25-((x)+(18-x)+y))}}} which simplifies to 7-y.
only mathematics: {{{22-((x)+(17-x)+(y))}}} which simplifies to 5-y.&lt;br&gt;
We now have expressions for the numbers of students that take each combination of courses:&lt;br&gt;&lt;pre&gt;
CMP     x
CM      y
CP      18-x
MP      17-x
C       7-y
M       5-y
P       x-10
-------------
total   37-y&lt;/pre&gt;
But the total number of students is 36; so y=1.&lt;br&gt;
That means 7-1=6 students take only chemistry and 5-1=4 take only mathematics.&lt;br&gt;
Then since 15 students take only one of the three courses, the number that take only physics is 5.  So x-10=5, making x=15.  So now&lt;br&gt;
x=15 take all three subjects
18-x=3 take only chemistry and physics
17-x=2 take only mathematics and physics&lt;br&gt;
And now we have all the numbers....&lt;br&gt;&lt;pre&gt;
CMP     15
CM       1
CP       3
MP       2
C        6
M        4
P        5
-------------
total   36&lt;/pre&gt;
ANSWERS:
(a) 15 students study all three courses
(b) 1 student studies only chemistry and mathematics&lt;br&gt;
-------------------------------------------&lt;br&gt;
Added after seeing the solution from the other tutor....&lt;br&gt;
My solution is NOT incorrect.  It satisfies all of the given conditions.&lt;br&gt;
It also has every student taking at least one of the three courses.  Since the problem doesn&#39;t say otherwise, I think that can be assumed.&lt;br&gt;
He came up with a solution to a different problem that satisfies all the given conditions and has 1 student not taking any of the classes....&lt;br&gt;



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