In this example, the prof states that "$Q \to R$ doesn't depend on the assumption $Q$ so he can discharge it, but without assumption $Q$, he couldn't have concluded with $Q\to R$ so the answer still depends on the assumption $Q$?
Doesn't a discharged assumption means you don't need that assumption and the proof still works??
Before that final inference you had a proof of $R$ from the (undischarged hypotheses) $P$, $Q$, and $(P\wedge Q)\to R$.
But after that final inference, the conclusion of the proof is no longer $R$, but $Q\to R$. If you think about what a conditional means, we never need to assume $Q$ in order to prove $Q\to R$, since $Q\to R$ will automatically hold whenever $Q$ is false.
Another way to think about it is that we don't need to assume anything about $Q$ to get to this conclusion, since the requirements about $Q$ are now explicitly coded in the formula $Q\to R$.
Finally, you can think of it this way: If $Q$ is false, then $Q\to R$ holds trivially. If $Q$ is true, then you can apply the subproof whose conclusion was $R$ in order to prove $R$. Thus $Q\to R$ holds (assuming the other undischarged hypotheses in the proof).