now, if we a "prime factoring" of each number, we get
204 = 2*2*3*17
120 = 2*2*2*3*5
168 = 2*2*2*3*7
now, notice the GCD for all three is that common bolded values, namely 2*2*3, or 12, so each room will house 12 participants,
now, how many rooms for all German? 204 ÷ 12, or 17 rooms.
how many rooms for all English ones? 120 ÷ 12, or 10 rooms.
how many rooms for the French ones? 168 ÷ 12, or 14 rooms.
so the total amount of rooms for all of them is 17 + 10 + 14.