nose count vs enumeration

nose count and enumeration both are nouns.

Word	Noun	Adjective	Verb	Adverb
nose count	Yes	No	No	No
enumeration	Yes	No	No	No

As nouns, enumeration is a hypernym of nose count; that is, enumeration is a word with a broader meaning than nose count:

Other hypernyms of nose count include count, counting, numeration, reckoning, tally.

nose count (noun)	enumeration (noun)
a periodic count of the population	the act of counting; reciting numbers in ascending order
	a numbered list