Learn Hardcore Japanese: tukue no ue no hon ga kaapetto ni otimasita. - The book on the desk fell onto the carpet.

机の上: The first の links 机 (desk) and 上 (top), giving “the top of the desk”.
上の本: The second の connects 上 (top / surface) to 本 (book), turning “top” into an adjective-like modifier: “the book of/on/at that top”.

So 机の上の本 is a compact noun phrase meaning “the book that is on top of the desk.”

Why is it 机の上の本 and not 机の上に本 or 机の上にの本?

You’re choosing between two different structures:

Location phrase + に + verb
- 机の上に本があります。
  “There is a book on the desk.”
  Here, 机の上に is an adverbial phrase (“on the desk”), and 本 is just the subject.
Noun phrase modifying 本
- 机の上の本
  “The book on the desk.”
  Here, 机の上の is directly modifying 本, forming one long noun phrase.

You cannot say 机の上にの本. You either do:

机の上に本がある (location phrase + verb), or
机の上の本がある (location phrase turned into a modifier with の).

机の上の本が落ちました。 literally feels like “the ‘on-top-of-the-desk’ book fell.”

Why is there no particle like に after 上 (why not 机の上にの本)? Doesn’t 上 need に?

上 only needs に when it is being used as a location phrase with a verb:

机の上に本があります。 – There is a book on the desk.
机の上に置きました。 – (I) put it on the desk.

In 机の上の本, 上 is behaving like a noun in a noun–noun chain:

机の上 – “the top of the desk”
机の上の本 – “the book (of) the top of the desk” → “the book on the desk”

There is no direct に because 机の上の as a whole is functioning as an adjective-like modifier for 本, not a separate “location + に” phrase with a verb.

What is the role of に in カーペットに落ちました?

Here, に marks the destination / target location of the action:

カーペットに落ちました。
“[It] fell onto / to the carpet.”

This is the same に used for destinations of movement:

学校に行きます。– I go to school.
ベッドに飛び込みました。– (I) jumped onto the bed.

So in this sentence:

本が – the book (subject)
カーペットに – onto the carpet (destination)
落ちました – fell

Could we use へ, で, or を instead of に with カーペット? What would change?

For this sentence, に is the natural and standard choice.

カーペットに落ちました。 – Fell onto the carpet. (destination)

Other particles:

へ – also marks direction (“toward”), but it’s less common with 落ちる.
カーペットへ落ちました is possible but sounds more like written style / narration and emphasizes direction rather than end-point.
で – marks the place where an action happens, not the destination:
カーペットで転びました。 – I fell on the carpet (the falling took place there).
With 落ちる, you’re normally interested in where it ends up, so に is better.
を – there is a pattern 場所を出る / 通る (leave/pass through a place), but カーペットを落ちました doesn’t work. You don’t “fall the carpet as a path”.

So, for “fell onto the carpet,” カーペットに落ちました is the correct, natural phrasing.

What is the difference between 落ちました and 落としました?

They are different verbs:

落ちる (おちる) – intransitive: “to fall” (by itself)
- 本が落ちました。 – The book fell.
落とす (おとす) – transitive: “to drop [something]” (someone causes it)
- 本を落としました。 – I/you/he dropped the book.

In the original sentence:

机の上の本がカーペットに落ちました。

we are describing the book falling by itself (or at least, without mentioning who caused it), so the intransitive 落ちました is used.

If you wanted to say “I dropped the book that was on the desk onto the carpet,” you’d use:

机の上の本をカーペットに落としました。

What does ました add to 落ち in 落ちました? Is this just past tense?

ました is the polite past form of the verb ending ます.

Dictionary form: 落ちる – to fall
Polite non-past: 落ちます – falls / will fall
Polite past: 落ちました – fell

So 落ちました simultaneously expresses:

past tense – the action already happened
politeness – respectful / neutral speech level

Casual past would be:

落ちた – “(it) fell.”
→ 机の上の本がカーペットに落ちた。 (casual)

If the English translation says “fell from the desk onto the carpet,” why is there no word like “from” (から) in the Japanese sentence?

The phrase 机の上の本 already tells you which book it is: “the book on the desk.” Japanese doesn’t have to explicitly say “from the desk” because it’s clear that if the book was on the desk and ended up on the carpet, it fell from there.

If you want to make “from” explicit, you can say:

机の上から本がカーペットに落ちました。
“From the top of the desk, a book fell onto the carpet.”

or more specific:

机の上の本が机の上からカーペットに落ちました。
(a bit redundant, but very explicit)

In the original, Japanese just uses the location in the noun phrase (机の上の本) and the destination with に, and leaves “from” to context.

Can I change the word order, like カーペットに机の上の本が落ちました? Is that still correct?

Yes. Japanese word order is flexible as long as you keep the correct particles:

机の上の本がカーペットに落ちました。
カーペットに机の上の本が落ちました。

Both are grammatically correct. The nuance:

Putting 机の上の本が first tends to focus on “the book on the desk”.
Putting カーペットに first can slightly emphasize the place where it ended up.

However, the most neutral and common order for this kind of simple sentence is the original:

[subject が] [destination に] [verb]
→ 机の上の本がカーペットに落ちました。

There is no “a” or “the” in the Japanese sentence. How do we know if it’s “a book” or “the book”?

Japanese does not have articles like “a” or “the”. Nouns are typically bare:

本 – can be “a book,” “the book,” or just “book(s).”

In context, several things help you interpret it:

The phrase 机の上の本 specifies which book: the one on the desk.
In English, this kind of specific noun phrase is usually translated with “the”:
→ “The book on the desk fell onto the carpet.”

If context made it clearly indefinite, you might translate it as “a book,” but with 机の上の本, it’s naturally understood as a particular, already-known or visually identifiable book.