An urn contains nine balls of which three are red, four are blue and two are green. Three balls are drawn at random without replacement from the urn. The probability that the three balls have different colour is

Options

A. `(2)/(23)`
B. `(1)/(3)`
C. `(2)/(7)`
D. `(1)/(21)`

Explanation

To solve the problem of finding the probability that three balls drawn from an urn containing red, blue, and green balls are of different colors, we can follow these steps:

### Step 1: Identify the total number of balls and their colors
The urn contains:
- 3 Red balls
- 4 Blue balls
- 2 Green balls

Total number of balls = 3 + 4 + 2 = 9 balls

### Step 2: Define the event of interest
Let event A be the event that the three balls drawn are of different colors. This means we need to draw:
- 1 Red ball
- 1 Blue ball
- 1 Green ball

### Step 3: Calculate the number of favorable outcomes
To find the number of ways to choose 1 ball of each color, we can use combinations:
- The number of ways to choose 1 Red ball from 3 Red balls = \( \binom{3}{1} = 3 \)
- The number of ways to choose 1 Blue ball from 4 Blue balls = \( \binom{4}{1} = 4 \)
- The number of ways to choose 1 Green ball from 2 Green balls = \( \binom{2}{1} = 2 \)

Now, we multiply these together to find the total number of favorable outcomes:
\[ \text{Favorable outcomes} = 3 \times 4 \times 2 = 24 \]

### Step 4: Calculate the total number of outcomes
Next, we need to calculate the total number of ways to choose any 3 balls from the 9 balls:
\[ \text{Total outcomes} = \binom{9}{3} = \frac{9!}{3!(9-3)!} = \frac{9 \times 8 \times 7}{3 \times 2 \times 1} = 84 \]

### Step 5: Calculate the probability
The probability that the three balls drawn are of different colors is given by the ratio of favorable outcomes to total outcomes:
\[ P(A) = \frac{\text{Favorable outcomes}}{\text{Total outcomes}} = \frac{24}{84} \]

Now, we simplify this fraction:
\[ P(A) = \frac{24 \div 12}{84 \div 12} = \frac{2}{7} \]

### Final Answer
Thus, the probability that the three balls drawn have different colors is \( \frac{2}{7} \).

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