Why is детей in the Gen. and not the Nom. in this у + Gen. + Nom. construction?

Question

Apologies for this basic question:

Why does 'детей' in 'сколько у тебя детей' appear in the gen. and not the nom. as in the usual у + gen. + nom. construction?
I wanted to ask this question as the preceding sentence read: 'у тебя есть дети' where the у + gen. + nom. seemed to work perfectly. Is it because

• (1) the 'existence' of the children is no longer in question;
• (2) that the noun [детей] must agree with the personal pronoun [тебе] in the genitive case; or
• (3) because we're now asking about the quantity of children?

Apologies, I just really want to know for sure. If anyone could explain exactly why it changes I'd be very grateful - thanks!

Answer 1

• "Сколько" – as a word denoting the quantity of something – takes a genitive noun by default.

• An associated genitive noun (like "детей") is often separated from "сколько" rather than being placed immediately following "сколько".

In one of the simplest examples, "How old are you?" – while it is perfectly acceptable to place "сколько" and the associated genitive noun "лет" side by side:

А сколько лет тебе?

Separated style – with some pronoun (тебе, у тебя, etc) interposed – is more commonly used:

А сколько тебе лет?

Сколько тебе было лет, когда умерла твоя мать?

Answer 2

Here, сколько means quantity. Let's count.

1 год, 2 года, 3 года, 4 года, 5 лет, 6... лет 1 ребёнок, 2-4 ребёнка, 5... детей.

Сколько лет, сколько детей - as in the biggest variant. Don't pay attention that 21 - again год, just remember, that сколько requires the same form as with 5, 6, 7, 8, 9 and 10.

The same case is with the adverbs много, мало, несколько.

Много людей, мало сил, несколько яблок.

So,

У тебя есть несколько яблок? Мы сделаем пирог.
У нас есть много друзей, и все они придут в гости.
У меня мало сил к вечеру, сделаю работу завтра.