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Though I've found one of the questions with the same topic as mine, I would desire the better elaborated explanation as it confuses me a lot.

According to that question, the answer was that when the word 'solution' is related to answer for social problems, then the preposition 'to' is the most suitable as well as commonly used, and when talking about math problems, for example we say "the solution of this equation", then 'of' fits.

Now my question is, is it grammatically correct to use 'of' for the former case?

To be more specific,

Kim Kardashian tried to find solutions [to/of] the problem that she and her friends had faced.

I'm aware that in this case we should prefer 'to'. Nevertheless, in this case can't I use 'of' at all? If I do it is it grammatically fine?

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This is one of those contexts where usage has changed significantly over time...

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Note that this chart represents written instances. The actual shift in real-world conversational contexts is much greater than implied above (because the written form usually lags behind spoken usage, and many written examples are simply referencing / repeating earlier instances anyway).

At the level of "formal syntax" I don't think there's a specific rule saying when you can reasonably refer to a solution for / of a problem, but I'm sure there are contexts where of is at least "okay". And undeniably, for (not of) is "acceptable" in OP's specific context (but to is definitely far more common today).

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