Rule of inference. Is the argument valid?

The argument is not valid.

The second, third, and fourth premises entail $neg p$ ... via modus tollens.

$$pto q, qto(kwedge t), neg(kwedge t)vdash neg p$$

So any interpretation which satisfies these three premises will also satisfy the first premise.

However, notice that their valuation is not dependent on the valuation of $r$.

Thus we may value $r$ as false in some interpretations where all four premises are satisfied.

Therefore $r$ is not entailed by these four premises.

This argument is invalid. Multiple applications of Modus Tollens on 2, 3, 4 leads to $neg p$.

Hence 1 always holds and no conclusion can be drawn on $r$.

Another way to show that this is invalid is via truth tables.

As long as $p=q=k=t=F$, $r$ can be $T$ or $F$.

The mistake in the question is either 1 should be $(p lor r)$ or $(neg p to r)$ (which are equivalent to each other.)