В наборе три чашки.

Because в here means in in the sense of location, and after в Russian normally uses the prepositional case for location.

• Base form: набор
• After в for location: в наборе

So В наборе means in the set.

Compare:

• в наборе = in the set
• в набор = into the set

In Russian, the verb to be is usually omitted in the present tense.

So instead of saying something literally like In the set there are three cups, Russian very naturally just says:

• В наборе три чашки.

This is one of the most common sentence patterns in Russian.

After the numerals 2, 3, 4, Russian uses a special noun form. For чашка, that form is чашки.

So you get:

• одна чашка = one cup
• три чашки = three cups
• пять чашек = five cups

This is a very important number pattern in Russian:

• 1 → singular
• 2, 3, 4 → the чашки form
• 5+ → the чашек form

After три, чашки is traditionally analyzed as genitive singular, even though for this noun it looks exactly the same as nominative plural.

That means:

• dictionary form: чашка
• genitive singular: чашки
• nominative plural: чашки

So the form looks familiar, but in три чашки it is the form required by the numeral три.

For a beginner, the most useful rule is simply:

• after 2, 3, 4, use the чашки form

Набор usually means a set, kit, or package of items that belong together.

So В наборе три чашки means the set includes three cups. It does not necessarily focus on physical location inside a box. It is more about what belongs to the set.

Depending on context, набор could be:

• a tea set
• a dish set
• a gift set
• a boxed set of items

Yes. Russian word order is flexible, but different orders change the emphasis.

• В наборе три чашки.
  Natural, neutral. It starts with the setting: in the set.

• Три чашки в наборе.
  Possible, but it sounds more marked. It might be used for contrast, like if you are comparing sets.

So the given word order is the most natural if you are simply describing the contents.

Because Russian uses different prepositions in ways that do not always match English exactly.

Here, в is used because the idea is within the set / as part of the set. Russian treats набор more like something that contains items conceptually.

So:

• в наборе = in the set / included in the set

Using на here would sound wrong for this meaning.

No. Russian has no articles.

So В наборе три чашки could mean:

• There are three cups in the set
• There are three cups in a set

Which one is meant depends on context.

Russian learners often have to get used to this: definiteness is usually understood from the situation, not from a word like the or a.

The stress is:

• в набОре три чАшки

A rough pronunciation guide:

• в = like v
• набОре = na-BO-rye
• три = tree
• чАшки = CHAsh-kee

The most important thing is to remember the stressed syllables:

• набОре
• чАшки

Yes. A common alternative is:

• В набор входят три чашки.

This means something like Three cups are included in the set.

Both are correct, but they feel slightly different:

• В наборе три чашки. = simple statement about contents
• В набор входят три чашки. = slightly more explicit, often used in product descriptions

The original sentence is shorter and very natural.