To solve this problem, we can use the concept of probability. The probability of an event happening is defined as the number of favorable outcomes divided by the number of possible outcomes.

Let's break down the problem step by step.

Step 1: Find the total number of balls in the bag.
There are 3 red balls, 4 green balls, and 3 yellow balls. So, the total number of balls in the bag is 3 + 4 + 3 = 10.

Step 2: Find the number of ways to choose 2 balls of different colors.
To find the number of ways to choose 2 balls of different colors, we need to consider all the possible combinations of different colors. We can choose one ball from the 3 red balls and one ball from the 4 green balls, or one ball from the 3 red balls and one ball from the 3 yellow balls, or one ball from the 4 green balls and one ball from the 3 yellow balls.

Number of ways to choose 2 balls of different colors = (3C1 * 4C1) + (3C1 * 3C1) + (4C1 * 3C1)
= (3 * 4) + (3 * 3) + (4 * 3)
= 12 + 9 + 12
= 33

Step 3: Find the total number of ways to choose 2 balls from the bag.
The total number of ways to choose 2 balls from the bag is given by the combination formula.

Total number of ways to choose 2 balls = 10C2
= (10 * 9) / (2 * 1)
= 45

Step 4: Calculate the probability.
The probability of drawing 2 balls of different colors is given by the number of ways to choose 2 balls of different colors divided by the total number of ways to choose 2 balls.

Probability = Number of ways to choose 2 balls of different colors / Total number of ways to choose 2 balls
= 33 / 45
= 11 / 15

Therefore, the probability that the two balls drawn are of different colors is 11/15, which corresponds to option C.