Learn Hardcore Japanese: ryou no sangai ni ryuugakusei ga iru. - There is an international student on the third floor of the dormitory.

に with いる／ある → location of existence
- 寮の三階に留学生がいる。
  = On the third floor of the dorm, an international student exists / is.
で with verbs of action → place where something is done
- 寮の三階で留学生が勉強している。
  = On the third floor of the dorm, an international student is studying.

Because いる is a verb of existence, 三階 takes に, not で.

Why is が used after 留学生? Why not は?

In sentences of existence like X に Y がいる／ある, が is the normal subject marker for Y (the thing that exists).

寮の三階に留学生がいる。
The new information you are presenting is 留学生 (an international student), so it is marked by が.

Using は would normally make 留学生 the topic rather than pure new information:

留学生は寮の三階にいる。
= As for the international student(s), (they) are on the third floor of the dorm.

So:

X に Y がいる → neutral “there is Y in X”, introducing Y as new information.
Y は X にいる → “as for Y, (they) are in X”, talking about Y as a known topic.

Can I say 留学生が寮の三階にいる instead? Is that different?

Yes, 留学生が寮の三階にいる。 is also correct.

Both:

寮の三階に留学生がいる。
留学生が寮の三階にいる。

are grammatically fine and usually mean the same thing in context.

Nuance / feel:

寮の三階に留学生がいる。
Slightly emphasizes the location first: “On the third floor of the dorm, there is an international student.”
留学生が寮の三階にいる。
Slightly emphasizes the person first: “The international student is on the third floor of the dorm.”

In everyday speech, both orders are natural; context and intonation give most of the nuance.

Why is いる used and not ある?

Japanese has two common “to be / there is” verbs:

いる – used for animate things: people, animals, some personified characters
ある – used for inanimate things: objects, places, abstract things

Because 留学生 is a person (animate), you must use いる:

寮の三階に留学生がいる。 ✅
寮の三階に留学生がある。 ❌ (unnatural)

By contrast:

寮の三階にトイレがある。
= There is a toilet on the third floor of the dorm.

What exactly does 寮の三階 mean, and what is の doing here?

寮の三階 literally is:

寮 – dormitory
の – “of” / possessive / belonging marker
三階 – third floor

So 寮の三階 = “the third floor of the dorm”.

Grammatically, の links nouns in a Noun A の Noun B pattern, where:

Noun B is the main thing (here, 三階)
Noun A gives extra information (here, that this floor belongs to or is part of the dorm, 寮)

So 寮の三階 = “the dorm’s third floor / the third floor of the dorm”.

Could I say 三階の寮 instead of 寮の三階?

No, that would mean something different.

寮の三階 – the third floor of the dorm (a floor inside a specific dorm)
三階の寮 – a dorm that is on the third floor (a dorm located on some building’s third floor)

In this context, you want the first meaning: a floor of the dorm building itself, so 寮の三階 is correct.

Does 留学生 here mean one student or several students?

留学生 by itself does not show singular or plural. It can mean:

“an international student”
“international students”

Which one it is depends on context (and sometimes on how the speaker continues, e.g., using 一人, 二人, etc.).

If you want to clearly say one international student, you can add a counter:

寮の三階に留学生が一人いる。
= There is one international student on the third floor of the dorm.

For more than one:

寮の三階に留学生が三人いる。
= There are three international students on the third floor of the dorm.

Why doesn’t the sentence use articles like “a” or “the” for 留学生?

Japanese does not have words that directly match English a / an / the. Nouns like 留学生 are neutral with respect to definiteness and number.

留学生 could be “a(n) international student”, “the international student”, or “international students”, depending on context.

You rely on:

previous context (has this person been mentioned?),
counters (like 一人, 二人),
and sometimes demonstratives (この, その, あの)

to show the kind of specificity that English articles express.

Can I drop any particles here in casual speech, like が or に?

In very casual spoken Japanese, が is often dropped, especially when the meaning is clear:

寮の三階に留学生がいる。 (full form)
寮の三階に留学生いる。 (casual; が omitted)

This is common and natural in conversation.

Dropping に is much less natural here, because に shows the location for いる:

寮の三階留学生いる。 – This can be heard, but sounds rough / telegraphic and is more like something you might see in notes or hear in very casual/fast speech.

For learners, it’s safest to keep both に and が until you are comfortable with when they can be omitted.

How would I make this sentence polite?

Use the polite form of いる, which is います:

寮の三階に留学生がいます。

That is the standard polite version.

If you turned it into a question:

寮の三階に留学生がいますか。
= Is there an international student on the third floor of the dorm?

Is there a general pattern I can remember for sentences like this?

Yes. A very common pattern for existence/location sentences in Japanese is:

Location + に + Thing/Person + が + いる／ある

Applied to your example:

寮の三階 (location)
に (location marker)
留学生 (person/thing)
が (subject marker)
いる (existence verb for people/animals)

So the template is:

X に Y がいる。 – There is Y in/at/on X.
X に Y がある。 – There is Y in/at/on X. (for inanimate Y)

Remembering this pattern will help you make many similar sentences.