I know that "the" is wrong here, but I can't explain why it's wrong to my friend here, and I have trouble formulating a google search to find some descriptive "rule" or something. Help please?
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Because "booth 24" and "room 56" are acting as proper nouns (names). Just like you would say let's meet at "ABC" , where "ABC" is the name of a restaurant.
I would not see booth 24 as proper noun, but booth 24 is so precise an indication that is unnecessary to use the.
It's like saying the date. You say:
Because what warrants the use of "the" here is that specific day of September.
In your example, what warrants the "the" is the specific number, not the fact that it's a room among other places. So you have two choices:
I hope this makes sense.
Edit: I guess time isn't a proper analogy but it carries the meaning across. Because when telling time, "September the 3rd" would have been perfectly acceptable too but "room the 24th" doesn't work.