A bag contains $2$ red roses, $4$ yellow roses and $6$ pink roses. Two roses are drawn at random. What is the probability that they are not of same color?

This problem can be easily solved with $$\frac{2C1 \cdot 4C1 + 4C1 \cdot 6C1 + 6C1 \cdot 2C1}{12C2} = \frac{2}{3}$$

But I want to do this using the ratio method and here is my attempt:
RY + YP + PR =
$$\frac{2}{12} \cdot \frac{4}{12} + \frac{4}{12} \cdot \frac{6}{12} + \frac{6}{12} \cdot \frac{2}{12} = \frac{11}{36}$$
which is way off, what am I doing wrong? and also how does one decide when to use the ratio method and when to use the Combinatorics method.

Thank you for any help.