pas de trois vs dancing

pas de trois and dancing both are nouns.

Word	Noun	Adjective	Verb	Adverb
pas de trois	Yes	No	No	No
dancing	Yes	No	No	No

As nouns, dancing is a hypernym of pas de trois; that is, dancing is a word with a broader meaning than pas de trois:

pas de trois: (ballet) a dance for three people
dancing: taking a series of rhythmical steps (and movements) in time to music

Other hypernyms of pas de trois include dance, saltation, terpsichore.

pas de trois (noun)	dancing (noun)
(ballet) a dance for three people	taking a series of rhythmical steps (and movements) in time to music