У кімнаті тринадцять стільців.

Because after у / в meaning in, Ukrainian uses the locative case for location.

• кімната = room, in the basic dictionary form
• у кімнаті = in the room

So кімнаті is the locative singular form of кімната.

Because after numbers like 5, 6, 7... and higher, Ukrainian normally uses the genitive plural form of the noun.

• стілець = chair
• стільці = chairs, nominative plural
• стільців = of chairs, genitive plural

So:

• один стілець = one chair
• два / три / чотири стільці = two / three / four chairs
• п’ять стільців = five chairs
• тринадцять стільців = thirteen chairs

That is why стільців is correct here.

In the present tense, Ukrainian often omits the verb to be.

So instead of saying something like There are thirteen chairs in the room, Ukrainian simply says:

У кімнаті тринадцять стільців.

Literally, it is closer to:

In the room thirteen chairs.

This is completely normal and natural in Ukrainian.

Ukrainian word order is more flexible than English. Starting with У кімнаті puts the location first:

У кімнаті тринадцять стільців.
= In the room, there are thirteen chairs.

This is a very natural way to present new information: first the place, then what is there.

You could also say:

Тринадцять стільців у кімнаті.

But that sounds different in emphasis and is less neutral in many contexts.

Both у and в can mean in. The choice is often about sound and ease of pronunciation, not meaning.

Ukrainian switches between them to avoid awkward sound combinations.

For example:

• У кімнаті sounds natural
• In other contexts, в might sound better

So here у and в would mean the same thing, but У кімнаті is the smoother choice.

Because 13 belongs to the group 11–14, and these numbers take the noun in the genitive plural.

So:

• одинадцять стільців = eleven chairs
• дванадцять стільців = twelve chairs
• тринадцять стільців = thirteen chairs
• чотирнадцять стільців = fourteen chairs

Even though три by itself would normally give три стільці, the number тринадцять behaves like 5+, not like 3.

More or less, yes:

• У = in
• кімнаті = the room / room in the locative
• тринадцять = thirteen
• стільців = chairs in the genitive plural

So the literal structure is:

In the room thirteen chairs.

That sounds incomplete in English, but in Ukrainian it is a normal full sentence meaning There are thirteen chairs in the room.

The dictionary forms are:

• кімната = room
• тринадцять = thirteen
• стілець = chair

In the sentence, two of them change form:

• кімната → кімнаті because of the locative after у
• стілець → стільців because it follows тринадцять

The number тринадцять stays the same here.

A rough pronunciation guide is:

U kim-NA-ti try-NAD-tsyat STI-ltsiv

A few useful notes:

• У sounds like oo in boot
• і sounds like ee in see
• ц sounds like ts
• ь in стільців softens the л

The main stresses are usually:

• кімна́ті
• трина́дцять
• стільці́в

So: У кімна́ті трина́дцять стільці́в.