У бібліотеці є двадцять один підручник.

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

• Dictionary form: бібліотека
• Locative singular: бібліотеці

So у бібліотеці means in the library. This is a very common pattern:

• у школі = in the school
• у кімнаті = in the room
• у бібліотеці = in the library

Both у and в can mean in. Ukrainian often chooses between them for euphony—whichever sounds smoother next to the surrounding sounds.

Here, у бібліотеці is very natural because it avoids a heavy consonant sequence at the start. But в бібліотеці is also possible in many contexts.

So the difference here is mostly about sound and flow, not meaning.

Є is the present-tense form used for there is / there are in existential sentences.

So in this sentence, it marks existence:

• У бібліотеці є двадцять один підручник.

In everyday Ukrainian, є is often omitted when the meaning is clear:

• У бібліотеці двадцять один підручник.

Both are possible. Including є makes the existence idea more explicit and is very normal.

Because numbers ending in 1 normally take a singular noun, as long as the number is not 11.

So:

• 21 підручник
• 31 підручник
• 41 підручник

But:

• 11 підручників

In this sentence, the last part of the numeral is один, so the noun stays singular:

• двадцять один підручник

Because один must agree with the noun in gender.

Підручник is a masculine noun, so the form is:

• один підручник

Compare:

• одна книга — feminine
• одне вікно — neuter

So if the noun changed, the form of one would change too:

• двадцять одна книга
• двадцять одне вікно

The noun form changes depending on the number.

With this noun:

• 21 підручник
• 22 підручники
• 25 підручників
• 11 підручників

A useful basic rule is:

• numbers ending in 1 (but not 11) → підручник
• numbers ending in 2, 3, 4 (but not 12, 13, 14) → підручники
• numbers ending in 5–9, 0, and 11–14 → підручників

This number pattern is one of the most important things to learn in Ukrainian.

Here it is nominative singular.

That is because один behaves like an adjective and agrees with the noun:

• один підручник

Since двадцять один ends in один, the whole phrase keeps the noun in the same singular form:

• двадцять один підручник

So even though the quantity is large, the grammar follows the final numeral один.

No. Ukrainian word order is fairly flexible.

This version is a very natural, neutral way to present the information:

• У бібліотеці є двадцять один підручник.

You could also hear:

• У бібліотеці двадцять один підручник.

Changing the order usually changes emphasis more than basic meaning. Putting у бібліотеці first is very common when you are saying what exists in a place.

Because Ukrainian has no articles.

Unlike English, Ukrainian does not normally use separate words for a/an or the. Whether something is definite or indefinite is usually understood from:

• context
• word order
• the situation
• extra words like цей if needed

So підручник can correspond to a textbook or the textbook, depending on context.

Not exactly.

• книга = book
• підручник = textbook, schoolbook, coursebook

So підручник is more specific: it is a book used for studying a subject.