You are right that 4 and 5 are incorrect. All the others are fine.
7 is incorrect, yes.
8 will be correct if you write it like this: I don't know how he solved the problem. (second verb is past tense)
"how" in this sentence is serving as an interrogative word to introduce an indirect question.
1. This is the house where he lives.
2. This is the house in which he lives.
3. This is the house that/which he lives in.
4. This is the house that he lives.
5. This is the house he lives.
6. This is the house he lives in.
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Among the 6 sentences, #4 and #5 are incorrect? Am I right? Are the others grammatical?
7. I don't know the way how he solve the problem.
8. I don't know how he solve the problem.
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#7 is wrong. #8 is right. We have to omit either 'the way' or 'how.' Then what is the part of speech of 'how' in Sentence 8? Is 'how' an interrogative adverb or a relative adverb?
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