<s>[INST] I don't know what is wrong with my graph for this problem - at most how many students take none of these courses? Is my graph wrong? If it is, what is the correct one? I used this graph to solve the problem and got a wrong answer. It is highly appreciated if one can show me the correct graph.

Of the 60 students at Hope Middle School, 40 take algebra, 45 take geometry, 48 take trigonometry. Furthermore, 22 take all three courses. At most how many students take none of these courses?


According to this graph, I have:
$$a+b=45-22$$
$$b+c=48-22$$
$$c+a=40-22$$
$$x=60-(40+45+48-a-b-c-22)$$
$x$ is many students taking none of these courses.
 [/INST] 
Total of all the classes
$$
a+b+c+2d+2e+2f=133-66=67\tag{1}
$$
Algebra
$$
a+d+e=40-22=18\tag{2}
$$
Trigonometry
$$
b+d+f=48-22=26\tag{3}
$$
Geometry
$$
c+e+f=45-22=23\tag{4}
$$
Number of students in one of these classes
$$
t=a+b+c+d+e+f+22\tag{5}
$$
Number of students not in any class
$$
\begin{align}
60-t
&=38-a-b-c-d-e-f\\
&=38-\frac{67+a+b+c}2\\
&=\frac{9-(a+b+c)}2\tag{6}
\end{align}
$$
To maximize $(6)$, we want to minimize a+b+c. We cannot have $a+b+c=0$ since $(1)$ would then imply that $2(d+e+f)=67$. Thus, the minimum would be $a+b+c=1$. There are three cases:
$a=1$: $b=c=0$, $d=10$, $e=7$, $f=16$
$b=1$: $a=c=0$, $d=10$, $e=8$, $f=15$
$c=1$: $a=b=0$, $d=11$, $e=7$, $f=15$

Each of these cases leaves us with $4$ students not taking any classes.
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