Which use of apostrophe is correct in the sentence? "Because of staff change's, the list of birthday's was distributed to all worker's in the business' three offices."

The correct choice is the use of "business'" because it correctly indicates possession when referring to something belonging to the business. The apostrophe after "business" signifies that whatever follows belongs to the business, which is appropriate in this context, as it is referencing the offices owned or operated by the business.

In contrast, other words in the sentence are not correctly formed with apostrophes. For instance, "change's" improperly suggests possession of the staff change rather than referring to the concept of a staff change itself. Similarly, "worker's" indicates possession belonging to a single worker rather than to multiple workers, which the context of the sentence implies. Lastly, "birthday's" incorrectly suggests a singular possession rather than referring to multiple birthdays. Thus, “business'” as a plural possessive form is the only instance that adheres to the rules of proper apostrophe usage in this sentence.