"He is running with the old gentlemen."
Translation:Er rennt mit den alten Herren.
AFAIK, undefined noun would be nouns with no article (der, ein, kein). In which case, a plural noun with no article does go with den.
I set up a small table for easier memory ^^ I'd recommend doing that too
That's a good idea, but I think the problem is that the sentence in the exercise does in fact use a definite article (mit den alten Herren - with the old gentlemen), so I still don't see why anything in this sentence would be called "undefined". I'm not sure if I understood what you meant by saying that "a plural noun with no article does go with den". "Den" (the) is an article, so a noun accompanied by "den" isn't a noun with no article.
I have no idea what "undefined noun" is supposed to mean in this context. Where did you read this?
However, "gentlemen" in the English sentence is plural, so you have to use a plural in the German translation, too: "mit den alten Herren" (dative plural).
The singular "with the old gentleman" would be "mit dem alten Herrn (!)" in standard German, since "Herr" is a weak noun. http://www.vistawide.com/german/grammar/german_nouns03.htm
"Männer" means "men"; "gentlemen" means "Herren". The basic meaning is the same, but there is a difference in the level of politeness. So I'm not sure if your solution should be accepted.
Also, it would have to be "mit den alten Männern". After the preposition "mit", you have to use the dative case, and "Männer" is plural. In the dative plural, almost all nouns add an -(e)n, so "die alten Männer" (nominative and accusative plural) becomes "mit den alten Männern" (dative plural).
After "mit", you always have to use the dative case. "Gentlemen" (Herren) is plural. The dative plural of the definite article ("the") is "den": mit den alten Herren (with the old gentlemen, plural).
"Dem" is dative masculine/neuter singular. Ex. mit dem alten Herrn (with the old gentleman, singular).
See this table:
I feel your pain, but speaking as one who has tried to argue with language, I can assure you that it is no use. Language does not respond to the arguments of mere mortals such as you and I.
I've found it useful to simply think of "den" (the accusative masculine definite article) as a different word entirely from "den" (the dative plural), just as "die" (the feminine singular nominative article) is a different word from "die" (the plural nominative article).
For what it's worth, English also has words that have multiple meanings, some apparently unrelated. For example, "die" in English means to expire; it can also refer to a tool used in manufacturing; and it once referred to one of a pair of dice (although that meaning has fallen out of common usage). And then, of course, there is the word "dye," which is indistinguishable from "die" in speech....