There are a couple of reasons that option A is wrong:
1. The rising of costs is just not a great way of putting that idea across, so A is out for awkwardness, given that we have a better option.

2. We have of data-processing operations just after costs. This is different from saying just the rising of costs. The problem here is that we'd normally put a the before costs if costs is followed by of data-processing operations.

That is, even if we had to go with rising, we'd not normally say
The rising of costs of X...

We'd actually say
The rising of the costs of X...

We need to remember that the verbal section asks us to choose the best of the five options given to us in any question. This doesn't mean that the rising of is always wrong. It's wrong in this context, but there are a lot of places where it'd be fine. For example:

The rising of the tide...
The rising of the tides...

The rising of the seas...
The rising of the waters...

The rising of the Moon...
The rising of the Sun...
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Hi AjiteshArun EducationAisle

Thank you all for your insightful points of view.

Nevertheless, I am still not able to understand the question yet. My problem here is that, even though I realize how quirky "The rising of costs" sounds, in my humble opinion, is the only option correctly addressing the subject of the sentence.

What is creating the opportunity? "The cost" or "The rising"? It is not logical that costs per se are creating any opportunity, therefore the subject of our sentence is "The rise" and only in option a) we would find "The rise" functioning as the subject.

Please, correct me if there is something wrong in my reasoning.

We should consider the rising cost to be one unit. That is, we're not looking at cost, but at rising cost. This is the same as restricting ourselves to a cost that is rising. The sentence says that such a cost has created a growing opportunity for independent companies.

In other words, if you asked these independent companies what has benefited them, they could say "the rising cost of data-processing operations at financial institutions (our clients)."

I'm not sure I've answered your question, but please do let me know if I missed something in your question. I'm also putting one more sentence (from an official question) down below. Let me know how this one sounds. That will tell us whether it's just one question that we're working on, or something more general.

Declining values for farm equipment and land are going to force many lenders to tighten or deny credit this spring. (from here)
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Correct BunuelWannabe.

Grammatically, the difference is:
(i) Rising cost: Rising is used as an adjective (participle), modifying cost. So, cost is the subject
(ii) The rising of costs: Rising is used as a noun (gerund, as you mentioned in your earlier post). So, rising is the subject

As mentioned in my earlier post, if the intent is to depict rising as a noun, I would have preferred a pure-play noun rise.

Meaning-wise, the difference is:
(i) Rising cost: Costs that are in the process of rising (as Ajitesh mentioned)
(ii) The rising of costs: Does not explicitly suggest that the process of rising is still underway. In fact, it suggests that the rise itself (and not the process of rising) is being discussed.

For this question, a case can be made for either (i) or (ii); however, also notice that (A) and (C) also differ in costs vs cost. Since we have cost in the non-underlined portion (lower cost), it would be preferable to go with cost in the underlined portion as well.
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The rising of costs of data processing operations at many financial institutions has created a growing opportunity for independent companies to provide these services more efficiently and at lower cost.

A. The rising of costs
B. Rising costs
C. The rising cost
D. Because the rising cost
E. Because of rising costs

First point, choice D and E are fragments; they are simply subordinate clauses without a main clause and therefore are out.

The verb is the singular 'has created' in the non-underlined part in tandem, the subject has to be singular; therefore, B is out.
Between A and C, the use of the term 'rising' as an adjective of the cost is much better than the rising of costs, a gerundial noun.
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