Problem:
A bag contains 5 blue, 7 black, and 3 brown balls. 3 balls are drawn randomly. What is the probability that the balls drawn contain balls of different colors?

Solution:
To find the probability of drawing balls of different colors, we need to consider all possible combinations of 3 balls that can be drawn from the given bag and count the number of combinations that have balls of different colors.

Total number of combinations:
To find the total number of combinations, we can use the concept of combinations. The total number of combinations of 3 balls drawn from a bag containing 15 balls can be calculated using the formula:

C(n, r) = n! / (r!(n-r)!)

where C(n, r) represents the number of combinations of n objects taken r at a time, and n! represents the factorial of n.

In this case, n = 15 (total number of balls in the bag) and r = 3 (number of balls drawn). Plugging these values into the formula, we get:

C(15, 3) = 15! / (3!(15-3)!)
= 15! / (3!12!)

Number of combinations with balls of different colors:
Now, we need to count the number of combinations that have balls of different colors.

- Case 1: 1 blue, 1 black, and 1 brown ball
The number of ways to choose 1 blue ball out of 5 is C(5, 1) = 5! / (1!(5-1)!) = 5.
Similarly, the number of ways to choose 1 black ball out of 7 is C(7, 1) = 7! / (1!(7-1)!) = 7.
And, the number of ways to choose 1 brown ball out of 3 is C(3, 1) = 3! / (1!(3-1)!) = 3.
Therefore, the total number of combinations with 1 blue, 1 black, and 1 brown ball is 5 * 7 * 3 = 105.

- Case 2: 1 black, 1 blue, and 1 brown ball
Similarly, the total number of combinations with 1 black, 1 blue, and 1 brown ball is also 105.

So, the total number of combinations with balls of different colors is 105 + 105 = 210.

Probability:
The probability of an event is given by the ratio of the number of favorable outcomes to the total number of possible outcomes.

In this case, the number of favorable outcomes (combinations with balls of different colors) is 210, and the total number of possible outcomes (total combinations) is C(15, 3) = 15! / (3!12!) = 455.

Therefore, the probability that the balls drawn contain balls of different colors is 210 / 455 = 3/13.

Answer:
The correct answer is option B) 3/13.