1. I was wondering, what's the best way to travel around Europe?

2. I was wondering, what's the best way to travel around Europe.

3. I was wondering what's the best way to travel around Europe.

4. I was wondering what's the best way to travel around Europe?
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Is #1 right? Which one is right? Do we have to use a comma? Do we have to put a full stop or a question mark at the end?

#1 is correct, yes.

You need both the comma after "wondering" and the question mark at the end.

All four variations of the question you provided are correct, but they differ in punctuation usage. Let's break them down:

1. "I was wondering, what's the best way to travel around Europe?" - This sentence uses a comma to separate the introductory phrase ("I was wondering") from the actual question. This is a common way to structure a question.

2. "I was wondering, what's the best way to travel around Europe." - This is the same as #1, but without a punctuation mark at the end. In this case, the question mark is missing, which makes the sentence feel incomplete.

3. "I was wondering what's the best way to travel around Europe." - This sentence omits the comma found in the previous versions. The question is still clearly expressed, but without the pause created by the comma.

4. "I was wondering what's the best way to travel around Europe?" - This sentence is the same as #3, but with a question mark at the end. Including the question mark signals that the sentence is a question, helping the reader or listener understand the intended meaning more quickly.

In summary, all four variations are grammatically correct, but #1 and #4 are typically preferred since they use appropriate punctuation to indicate a question.