For now, let's consider that there are an infinite number of balls of each colour. So, we have to find the number of ways the girl can choose a total of 4 balls.
Think of colours as boxes, and we have 4 items to place in 3 different boxes. Visualise items as stars and boxes by the separators
One combination is: * | * | * * (1 red, 1 green, and 2 blue balls)
or: * | | * * * (1 red, 0 green, and 3 blue balls)
So, now we have to arrange 4 stars in 4 + 3 - 1 places (3 - 1 are the separators)
i.e. C(4 + 3 - 1, 4) = C(6, 4) = 15
Now, remove those cases when she chooses 4 red or 4 green balls.
So, the answer should be 15 - 2 = 13.