Why can't I use "appeared in our home" here?

[TheLearner]

Thanks, Mr. Micawber. But, would it be proper if the phrase "...who we haven't seen in a long time.." will be "...who we haven't seen for a long time.."?

[Mister Micawber]

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Yes. And the more formal version is 'whom we haven't seen in/for a long time'.
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Native English teacher at Mister Micawber's

[Don Geraldos]

"Grammatically, you can say 'came to our home' but I haven't heard any native speakers ever say it like that."

There a 55 million 'native speakers' in the UK, plus a few million more in Australia, South Africa and possibly Canada, who would find the answer 'to our home' much more natural than 'came by' - unless of course you mean they 'came by car', 'came by taxi' - or 'came [and passed] by'.

I think these two distractors are not sufficiently distinctive: I made the mistake, as you call it, of selecting 'came to our home'.