Q1 Are the following 2 sentences correct?
A: Mr. Brown came to see you during your absence.
B: Mr. Brown came to see you in your absence.
Q2 Are the following 2 sentences correct?
C: She helped her parents with their job during the summer vacation.
D: She helped her parents with their job in the summer vacation.
Q3 Are the following 2 sentences correct?
E: I went to the museum during my stay there.
F: I went to the museum in my stay there.
Q4 What's the difference between "during" and "in"?
B: Mr. Brown came to see you in your absence. ... when you were not present, in that situation when you were not present. Shows what happens in a certain state/situation, at a certain moment/snapshot in time.
Q2, Q3. Use only during.
in your absence is an idiomatic combination. If it weren't for the fact that it's a fixed expression, in wouldn't be a good choice there either.
It seems to me that we use in with a period of time (in a week, in an hour) to say how long it takes to do something. (I can do that job in two days. I solved the problem in two minutes.) But we don't normally use in with the entities themselves (vacation, stay) that take place over a period of time.
It happened [during / *in] [my summer vacation / my stay in London].
On the other hand, there are a lot of expressions that can use in as well as during. Many of them have to be learned as separate expressions. (during the summer / in the summer) Personally, I often prefer during in these situations.
It happened [during / in] [the war / his lifetime].
People are waiting to help.
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