On a question about 3 chairs, I was marked wrong for typing 椅子三つ instead of 三つの椅子. Aren't they both the same? Also, I thought the more typical syntax was to name the item first, followed by the number.
Regardless, if both are acceptable, neither should be marked wrong.
From what I've tried, the course accepts:
三つ(の)椅子があります。- Speaking of the three chairs, they are there. / There are three chairs.
椅子が三つあります。- Speaking of the chairs, there are three. / There are three chairs.
Emphasis is in the italics.
椅子三つ is something like "chairs three" (and 椅子の三つ would be chairs of three), doesn't really make sense.
Modifying part (how many?) comes before the modified part, just as adding adjectives to nouns. You would say "white cat/白い猫), not "cat white".