pas de quatre vs dancing

pas de quatre and dancing both are nouns.

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

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

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

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

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