The number of ways of choosing with parameters 
At our disposal is a collection of $10$ red, $11$ blue and $12$ yellow fabrics. (each fabric is unique) In how many ways can we choose $4$ different fabrics if we want at least one fabric of each of the three colors?

My solution was since the first fabric chosen must be red, there are $10$ options for it. Then the next fabric must be blue, which has $11$ options. The third fabric is yellow, with $12$ options, and the last fabric can be any of the colors, provided that it has not already been chosen, so there are $(9+10+11-3)= 30$  ways to choose the last one, making the total number of choices $9\cdot 10\cdot 11\cdot 30$.
My professor said that I needed to divide that by $2$ to get the right answer, but I just don't understand why. Any help would be much appreciated!

The total number of ways is $\binom {10} {2} \binom {11} {1} \binom {12} {1} + \binom {10} {1} \binom {11} {2} \binom {12} {1} + \binom {10} {1} \binom {11} {1} \binom {12} {2} = 19800.$
