У коробці чотирнадцять книг.

Because after у / в when you mean location, Ukrainian normally uses the locative case.

• коробка = box (dictionary form, nominative)
• у коробці = in the box (locative singular)

So the ending changes because the noun is doing a different grammatical job.

A very rough pattern is:

• nominative: коробка
• locative: у коробці

This is one of the first case changes learners meet in Ukrainian.

Because after чотирнадцять (fourteen), Ukrainian uses the genitive plural form of the noun.

Here is the important pattern:

• 1 книга
• 2, 3, 4 книги
• 5+ книг

But there is a special rule: numbers ending in 11, 12, 13, 14 also use genitive plural, so:

• 11 книг
• 12 книг
• 13 книг
• 14 книг

That is why чотирнадцять книг is correct.

It is genitive plural.

The dictionary form is:

• книга = book

The plural forms you are most likely to compare are:

• книги = nominative plural / sometimes also genitive singular depending on context
• книг = genitive plural

In this sentence, книг is required because of the numeral чотирнадцять.

Because Ukrainian often leaves out the present-tense form of to be.

So:

• У коробці чотирнадцять книг. literally feels like In the box fourteen books.

But the natural English meaning is:

• There are fourteen books in the box.

You can also say:

• У коробці є чотирнадцять книг.

Here є adds an explicit idea of existence, like there is / there are. In many situations, though, Ukrainian simply omits it.

Yes. In this sentence, у and в both mean in.

The choice is often about sound and flow, not meaning. Ukrainian frequently switches between у and в to avoid awkward sound combinations.

So both are possible:

• У коробці чотирнадцять книг.
• В коробці чотирнадцять книг.

A speaker may choose whichever sounds smoother in context.

Because Ukrainian has no articles.

English says things like:

• in the box
• fourteen books
• a book
• the book

Ukrainian does not have separate words for a/an or the. Instead, definiteness is understood from context.

So:

• У коробці can mean in the box
• книг can mean books

The exact English article depends on the situation, not on a word in the Ukrainian sentence.

No, Ukrainian word order is fairly flexible.

У коробці чотирнадцять книг is natural and puts the location first, something like:

• In the box, there are fourteen books.

You could also rearrange the sentence, but the emphasis changes. For example:

• Чотирнадцять книг у коробці.
  This puts more focus on fourteen books.

The original sentence is a very normal way to present the information.

Yes, absolutely.

Both are grammatical:

• У коробці чотирнадцять книг.
• У коробці є чотирнадцять книг.

The version without є is very common and natural. The version with є can sound a bit more explicitly existential, like stressing that the books are present there.

In many beginner contexts, it is useful to recognize both.

A useful breakdown is:

• У = in
• коробці = box in the locative case, so in the box
• чотирнадцять = fourteen
• книг = books in the genitive plural

So a very literal gloss would be:

• In the box fourteen books

And the natural English translation is:

• There are fourteen books in the box.