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None of them is "wrong", nor really "different" in meaning, but 1(c) and 2(a) are the preferred expressions (in AmE).

I can't explain why. But here is some confirmation.

https://books.google.com/ngrams/graph?content=opened+the+book+at+%C2%A0page%2C+opened+the+book+on+page%2C+opened+the+book+to+page&case_insensitive=on&year_start=1800&year_end=2000&corpus=17&smoothing=3&share=&direct_url=t1%3B%2Copened%20the%20book%20at%20page%3B%2Cc0%3B.t1%3B%2Copened%20the%20book%20to%20page%3B%2Cc0

https://books.google.com/ngrams/graph?content=book+fell+open+at%2C+book+fell+open+on%2Cbook+fell+open+to&case_insensitive=on&year_start=1800&year_end=2000&corpus=18&smoothing=3&share=&direct_url=t1%3B%2Cbook%20fell%20open%20at%3B%2Cc0