Translate into FOL I have to translate the following statement:
$$"\text{Every teacher feels happy if all his/her students surpass their subject. }"$$
Where, 
$P(x) := \text{x is a professor}$
$F(x) := \text{x feels happy}$
$S(x, y) := \text{x surpass the subject imparted by y}$
$A(x, y) := \text{x is student of y}$
$$$$
I have asked three different students and neither of us have the same answer, the four options are:
a) $\forall x\forall y\left((P(x)\wedge A(y,x)\wedge S(y,x))\rightarrow F(x)\right)$
b) $\forall x\forall y\left((P(x)\wedge (A(y,x)\rightarrow  S(y,x)) \rightarrow F(x))\right)$
c) $\forall x\left((P(x)\wedge \forall y(A(y,x)\rightarrow  S(y,x)) \rightarrow F(x))\right)$
d) $\forall x\left(\forall y(P(x)\wedge A(y,x)\wedge S(y,x))\rightarrow F(x)\right)$
I think the correct is $\textbf{c)}$, am I correct?
 A: Yes, (c) is the correct answer. It says, "for every teacher $x$, if every student of $x$'s surpasses $x$'s subject, then $x$ is happy."
Since this is logic, we should also correct a minor error in the statement: it should be, "Every teacher feels happy if all their students surpass their subject." (Or, "every male professor feels happy if all his students surpass his subject.")
For reference, here is why the other options are wrong:


*

*(a) says "for every professor and student, if $x$ is a professor and $y$ is a student of $x$'s and surpasses $x$'s subject, then $x$ is happy." This is logically equivalent to "Any professor with at least one student who surpasses their subject is happy", which is not saying the same thing.

*(b) is quite wrong if you look where the parentheses are nested; it is logically equivalent to "Any professor with at least one person who isn't their student but does surpass their subject is happy."

*(d) is the closest other than (c) to being correct, because it gets the placement of $\forall y$ at least correct. After moving "$P(x)$" out of the inner quantification, it says, "For every professor $x$, if all people are $x$'s student and surpass $x$'s subject, then $x$ is happy."
