Line 1: The line must be made of a 2 and a 3 with either one first, so 2 orderings. There are 4 choices for the 2 and 3 choices for the 3 thus, 2x4x3 = 24 distinct lines.
Line 2: The line must be made of two 2's and a 3 and the 3 can be first, second, or third, so 3 orderings. Since we can't use the same words from the first line, there are 3x2 choices for the 2's and 2 choices for the 3, thus, 3x3x2x2 = 36 distinct lines.
Line 3: Like the first line, we must use a 2 and a 3 in some order, so 2 orderings. There is only 1 choice left for each of the 2 and the 3 thus, 2x1x1 = 2 distinct lines.
Putting all the options together, we get:
24x36x2 = 1728