Hun troede ikke, at der var nogen chance for at få plads, men der var heldigvis et ledigt bord.

Here der is the Danish existential there, like English there was / there were.

So:

• at der var nogen chance = that there was any chance
• der var heldigvis et ledigt bord = there was fortunately a free table

This der does not mean a physical location like there over there. It is just used to introduce the existence of something.

A very common Danish pattern is:

• der er ... = there is/are ...
• der var ... = there was/were ...

For example:

• Der er en bog på bordet. = There is a book on the table.
• Der var mange mennesker. = There were many people.

Because Danish main clauses usually follow the V2 rule: the finite verb comes in the second position.

In this sentence:

• Hun = subject
• troede = finite verb
• ikke = negation

So the normal order is:

• Hun troede ikke ...

not:

• Hun ikke troede ...

This is similar to the basic Danish pattern:

• Jeg kommer ikke.
• Han ved ikke svaret.

The negation ikke normally comes after the finite verb in a main clause.

Troede is the past tense of tro, and it often means thought in the sense of believed / assumed.

So:

• Hun troede ikke ... = She didn’t think / believe ...

This is different from tænkte, which is more about the act of thinking.

A useful contrast:

• Jeg troede, han var hjemme. = I thought/believed he was at home.
• Jeg tænkte på ham. = I was thinking about him.

So in your sentence, troede is the natural choice because it refers to what she believed to be the case.

In Danish, nogen often means any, especially after negation.

So:

• ikke ... nogen chance = not ... any chance

This works very much like English:

• She didn’t have any chance
• Hun havde ikke nogen chance

Compare:

• ingen chance = no chance
• ikke nogen chance = not any chance

In many contexts, both are possible, though the phrasing is slightly different:

• Der var ingen chance. = There was no chance.
• Der var ikke nogen chance. = There wasn’t any chance.

Both are natural Danish.

It literally means something like to get room/space, but in context it usually means to get a place, to be seated, or to get in.

So:

• chance for at få plads = chance of getting a place / getting seated

The noun plads is very flexible in Danish. It can mean:

• space
• room
• a place
• a seat/spot

In a restaurant context, få plads often means there is room for you, or that you can get seated.

Not quite.

• plads = room/space/place
• bord = table

So the first part is more general:

• der var nogen chance for at få plads = there was any chance of getting a place / being seated

Then the second part gives the reason:

• der var heldigvis et ledigt bord = there was fortunately a free table

So the logic is:

She thought there was no chance of getting seated, but luckily there actually was a table available.

Because bord is a neuter noun in Danish:

• et bord = a table

In Danish, adjectives agree with the gender and form of the noun. With a singular indefinite et-word, the adjective usually takes -t:

• et ledigt bord
• et stort hus
• et lille barn

Compare with common gender en-words:

• en ledig stol
• en stor bil
• en lille by

So ledig becomes ledigt because bord is an et-word.

Ledig here means free, available, or unoccupied.

So:

• et ledigt bord = a free/available table

It is not exactly the same as tom (empty).

A useful contrast:

• et ledigt bord = a table that is available for customers
• et tomt bord = an empty table

In many restaurant situations those ideas overlap, but ledigt focuses more on availability, while tomt focuses more on the fact that no one is sitting there.

Heldigvis means fortunately / luckily.

In this sentence:

• men der var heldigvis et ledigt bord

the adverb comes after the finite verb var, which is a very common position in Danish main clauses.

So the structure is:

• der = subject-like element
• var = finite verb
• heldigvis = sentence adverb
• et ledigt bord = noun phrase

This placement sounds natural and neutral.

You could also sometimes move adverbs for emphasis, but the sentence as given is standard and idiomatic.

Because at der var nogen chance for at få plads is a subordinate clause introduced by at (that).

So:

• Hun troede ikke, at der var ...

means:

• She didn’t think that there was ...

In Danish writing, many people put a comma before a subordinate clause like this. You also have a comma before men, which joins two main clauses.

So the punctuation helps separate:

1. the main clause: Hun troede ikke
2. the subordinate clause: at der var nogen chance ...
3. the second main clause after men

No, the two at’s have different functions.

1. at der var nogen chance ...

   • Here at means that
   • It introduces a subordinate clause

2. for at få plads

   • Here at is the infinitive marker, like English to
   • få = get
   • at få = to get

So in the sentence:

• first at = that
• second at = to

This is very common in Danish, and the meaning depends on the structure around it.

Yes, chance is a normal and common Danish word.

It means:

• chance
• possibility
• opportunity

In everyday Danish, you will often hear:

• en chance = a chance
• ingen chance = no chance
• en god chance = a good chance

In this sentence, nogen chance means any chance.

So even though it looks familiar to English speakers, it is fully natural Danish vocabulary.

Yes, mulighed is possible in many contexts, but the nuance is a little different.

• chance often feels a bit more like odds or likelihood
• mulighed often feels more like possibility

So:

• der var nogen chance for at få plads = there was any chance of getting a place
• der var nogen mulighed for at få plads = there was any possibility of getting a place

Both can work, but chance sounds very natural here, especially if someone doubts whether they will be able to get a table.

It has two main parts joined by men (but):

1. Hun troede ikke, at der var nogen chance for at få plads
2. men der var heldigvis et ledigt bord

So the structure is:

• Main clause: Hun troede ikke
• Subordinate clause: at der var nogen chance for at få plads
• Coordinating conjunction: men
• Second main clause: der var heldigvis et ledigt bord

This is a very common Danish sentence pattern: a statement, followed by a contrasting statement introduced by men.