what is the difference?

English is an endlessly varied language in the forms of expression which can be considered "correct." Each sentence expresses the idea correctly, including teh last sentence.

no... what I don't understand is: if the structure with would like is would like +noun/gerund, then how come this sentence is possible:

I would like to live on th31st floor.

There is no rule that "would like" must be followed by a gerund.

I like to swim.

I like swimming.

I would like to sing.

I would like singing.

Note that they have slightly different meanings.

e.g. "I like to swim" vs. "I like swimming."

Scenario:

Able and Bob are watching the olympic swimming events.

One says, "I like swimming" and the other says "I like to swim".

In this scenario one has never swam, who said what?

"Swimming" never swam. Why? Cause "swimming" can describe the general activity, while the simple infinitive can't.

please explain gerunds further....greatest thanks...

Gerunds, if I remember correctly, are a sort of nouns that are derived from verbs.

"Running" is a gerund. Because 'run' is a verb, but if I say "running is fun" it becomes a noun.