The example sentence correctly recognizes that the two clauses are independent, but in attempting to connect them into one compound sentence the author overzealously includes an unnecessary co-ordinating conjunction after the semicolon. This is overkill! Either a comma with an appropriate coordinating conjunction OR a semicolon with no accompanying conjunction can be used to punctuate this sentence. Deleting "but" or replacing the semicolon with a comma would fix this sentence.

Semicolons must follow independent clauses. Independent clauses must include all of the necessary components of a complete basic grammatical sentence. In the example sentence the semicolon interrupts the independent clause "I don't believe that you did what you said you did." The rest of the sentence is correct as written, as a comma and coordinating conjunction are used to connect the two independent clauses. Semicolons should never interrupt any clause.

The sentence is actually a run on sentence, with two complete sentences being jammed together to make a grammatically incorrect sentence. Such compound sentences need to be separated either by a comma and a conjunction, or a semicolon. This makes "Fewer and fewer people signed up for the program each year, and eventually it had to be cancelled," the correct answer choice.

Because “If you want to succeed in Dr. Henderson’s course, you need to do three things” is an independent clause (a complete sentence that stands on its own), the list that follows it must be preceded by a colon. (Semicolons are used to separate two independent clauses, not an independent clause and a list.) In this sentence, the commas are all used correctly.

Because “have your mother cut your hair or go to a salon” isn’t a list of three or more items, no colon is necessary to introduce it. To correctly punctuate the sentence, you would instead use a comma between “do” and “have.” The question mark and other comma are correctly placed in the sentence.

Because the first sentence isn’t really introducing the second sentence but is rather a separate thought, a colon isn’t the correct punctuation mark. Instead, a semicolon should be used to separate these two independent clauses.

This sentence has no error. The part of the sentence preceding the list is an independent clause (a complete sentence that can stand on its own), so the list is correctly introduced with a colon. Each item in the list is correctly separated with a comma, and the introductory clause (“The way I see it”) is correctly set off from the main clause (“the animal shelter has three options”) with another comma.

Here, the colon is incorrectly splitting up a clause. You wouldn’t say “we need so much help: decorating.” Instead, you’d say “we need so much help decorating.” For the same reason, you also wouldn’t break up the longer sentence, even though it ends with a list. Elsewhere, the commas are correctly separating the items in the list and separating the main clause (“It’s a pity you can’t come to the party”) from the dependent clause (“since we need so much help decorating, setting up seating, and preparing food.”)

Because “I have no idea if she’d rather” isn’t an independent clause (a complete sentence that can stand on its own), the list that follows it should not be separated with a colon or with any other punctuation. You would only use a colon to introduce the list if the part of the sentence before the list was an independent clause. The correct sentence is as follows: "I have no idea if she’d rather have dinner at an elegant restaurant, go to a comedy club, or compete in a trivia challenge to celebrate our anniversary."

Here, the colon is the correct punctuation to separate the first part of the sentence (an independent clause) from the second part of the sentence (a three-item list); however, the "His" that immediately follows the colon should not be capitalized.