I agree with you that the question is ambiguous, and also that the most sensible answer would be C. However, one could make a more or less reasonable argument in favor of several other answers, too.
a. The common ancestor of whales and fish possessed genes for fins.
Technically, this statement is true. At least some of the fins of whales and fish are even distantly homologous, even though the lineages leading to tetrapods (including whales) and to ray-finned fishes diverged quite early in their evolutionary history.
(Specifically, the flippers of whales are modified tetrapod forelimbs, which are homologous to the pectoral fins of ray-finned fishes. Whether one should also consider the tails fins of whales and fish to be homologous is a bit more debatable: the actual fin structures are very different, and presumably share few if any developmental pathways, but the tail and the spinal column which they attach to and are powered by is clearly a shared feature of both groups.)
However, while shared evolutionary history may explain why fish and whales have similarly placed flippers / pectoral fins, and why they both have a flexible spine that supports undulatory swimming with the help of a tail fin, it does not explain the convergent evolution of whales' forelimbs (which were formerly adapted for walking on land) into fin-like flippers. So in that sense, while perhaps partially correct, this answer is also quite incomplete.
b. Whales and fish possess the same mutations in their genomes.
It's hard to even tell what this statement means. Insofar as it's a restatement of the fact that whales and fish share some of their evolutionary history (up to the divergence of the bone-finned and ray-finned fishes), it's no more or less correct than statement A above.
However, I suspect that the intended meaning of this statement is that whales and (ray-finned) fish would've separately evolved the same "fin-making" mutations after their lineages diverges, which is both clearly incorrect and also, even prima facie, statistically very unlikely. So I would rule this one out.
c. Fins evolved in whales and fish because they both use them to swim in water.
This statement is written using teleological language, which some may consider misleading, as it could be misinterpreted as implying that evolution was a directed process: it could be interpreted as saying that whales needed to be able to swim, so they chose to evolve fins (or that some guiding consciousness granted them fins). This is, of course, incorrect — or, at least, we have no scientific evidence of there being any such conscious force guiding evolution, or of any organism (with the arguable exception of humans) being able to deliberately direct their own evolution.
However, it's quite possible to interpret this statement in a way that makes it perfectly correct: as both whales and fish move around by swimming, possessing fins (or something similar to fins) is an advantageous trait for them, and has thus been favored by natural selection. If pressed, I would thus choose this one as the most correct answer out of the given options.
In particular, note that the argument made by AliceD that "they have to develop fins first before they can use them" is not really true: the ancestors of both whales and fish were swimming long before they evolved fins, and thus the selection pressure in favor of having fins (or fin-like structures) was already present before the fins themselves evolved.
In particular, the first aquatic ancestors of whales are believed to have swum much like modern-day otters and seals, using their flipper-like limbs as makeshift fins. While there's no discrete transition point where one could definitely say that whale limbs turned into proper fins, it is clear that the fins, and their direct evolutionary predecessors, were being actively used for swimming, and that this fact directly drove their evolution into the actual fins that modern whales have.
d. Fins evolved in whales and fish because of different mutations that occurred in their genomes.
While technically correct, this statement is so vague that it's all but meaningless. All evolution is driven by mutations, and those mutations, being basically random, are rarely if ever the same in any two lineages.
Certainly, it would be silly to claim that "different mutations occurring in their genomes" is an explanation of why whales and fish evolved fins: humans also constantly have different mutations occurring in our genomes, but we clearly don't seem to be evolving fins. Neither are pigs or bumblebees or sunflowers, even though all of those species (and, of course, all others too) are also constantly experiencing different mutations in their genomes.
Really, this answer reminds me of the story about a person in a car who gets lost and stops to ask a passerby where they are; the passerby, after a moment's thought, replies "you're in your car." While technically correct, that answer is absolutely useless — and so is this one, too.
e. Fins evolved in whales and fish because their most recent common ancestor swam in water.
Again, this answer could be interpreted as basically a restatement of answer A, and thus as being partially correct. However, it completely ignores the convergent selection pressure described in answer C, which is responsible for the more recent evolution of whale flippers into something resembling fish fins more than, say, human hands or the feet of hippopotamuses (currently believed to be the closest living relatives of whales and other cetaceans). Thus, I would not pick this answer, for the same reason that I wouldn't pick answer A.