Он ждал чьего-нибудь ответа.

Questions & Answers about Он ждал чьего-нибудь ответа.

Why is ответа in the genitive case instead of the accusative?
The verb ждать (to wait) usually takes the genitive case when you are waiting for something abstract, indefinite, or when you aren't sure if it will actually arrive. Since he is waiting for anybody's answer (which is indefinite), ответ becomes ответа. If he were waiting for a specific, tangible person or thing, it would take the accusative (for example, Он ждал сестру).
How do you decline a word like чей-нибудь?
The suffix -нибудь never changes. You only decline the first part, чей (whose), exactly as you would if it were alone. Because ответа requires the genitive case to match it, чей takes its masculine genitive form, чьего. Put them back together with the hyphen, and you get чьего-нибудь.
The lesson is about the prefix кое-. Could we use кое-чьего here instead?
Yes, but it changes the meaning! The prefix кое- is used when the speaker has a specific person in mind but chooses not to name them. Он ждал кое-чьего ответа would mean 'He was waiting for a certain someone's answer.' By using the suffix -нибудь instead, it shows he was waiting for anybody's answer and didn't care who it came from.
Why is it чьего-нибудь and not кого-нибудь?
Кого-нибудь is the genitive form of кто-нибудь (anybody). Чьего-нибудь is the genitive form of чей-нибудь (anybody's). We need the possessive form here because it's 'the answer of anybody' (anybody's answer), so we must use the possessive pronoun чей as the base.

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