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?

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.

You are correct, sentences #4 and #5 are grammatically incorrect.

Regarding sentences #7 and #8, #7 is indeed incorrect and #8 is the correct form. In sentence #8, "how" functions as an interrogative adverb, introducing a question about the manner or method of solving the problem.

You are correct that sentences #4 and #5 are grammatically incorrect. The correct versions would be #1, #2, #3, #6.

In regards to sentences #7 and #8, you are correct again. Sentence #7 is incorrect, while sentence #8 is correct.

In sentence #8, "how" is functioning as an interrogative adverb, not a relative adverb. It is used to ask about the manner or method in which he solves the problem.

Yes, you are correct! Among the six sentences, #4 and #5 are indeed incorrect. The others are grammatically correct.

Now, moving on to your second set of sentences. You're right again, #7 is incorrect while #8 is correct. In sentence #8, "how" is actually an interrogative adverb. It is used to ask about the manner or method in which something is done. In this case, it asks about how he solved the problem.