In a bag there are 16 balls of three different colors i.e red ,blue and green .Number of red and blue balls is 9 and difference between red and green ball is 4 then find the probability of getting a ball of each color if three balls are picked at random ?

Options

A. `(5)/(28)`
B. `(4)/(27)`
C. `(7)/(36)`
D. `(9)/(40)`

Answer

To solve the problem step-by-step, we will follow these steps:

### Step 1: Determine the total number of balls and their colors
We know that there are a total of 16 balls consisting of three colors: red, blue, and green.

### Step 2: Set up the equations based on the problem
1. Let the number of red balls be \( R \).
2. Let the number of blue balls be \( B \).
3. Let the number of green balls be \( G \).

From the problem, we have:
- \( R + B + G = 16 \) (total number of balls)
- \( R + B = 9 \) (the number of red and blue balls combined)
- The difference between the number of red and green balls is 4, which gives us two cases:
- \( R - G = 4 \) or
- \( G - R = 4 \)

### Step 3: Solve for the number of balls of each color
Using the equations:
1. From \( R + B = 9 \), we can express \( B \) as \( B = 9 - R \).
2. Substitute \( B \) into the total equation:
\[ R + (9 - R) + G = 16 \implies G = 16 - 9 = 7 \]
3. Now substitute \( G \) into the difference equation:
\[ R - 7 = 4 \implies R = 11 \quad \text{(not possible since total is 16)} \]
or
\[ 7 - R = 4 \implies R = 3 \]
4. Now substitute \( R \) back to find \( B \):
\[ B = 9 - R = 9 - 3 = 6 \]

### Step 4: Summarize the number of balls
Now we have:
- Red balls \( R = 3 \)
- Blue balls \( B = 6 \)
- Green balls \( G = 7 \)

### Step 5: Calculate the probability of picking one ball of each color
To find the probability of picking one ball of each color when three balls are picked at random, we can use the combination formula.

1. The number of ways to choose 1 red ball from 3 is \( \binom{3}{1} = 3 \).
2. The number of ways to choose 1 blue ball from 6 is \( \binom{6}{1} = 6 \).
3. The number of ways to choose 1 green ball from 7 is \( \binom{7}{1} = 7 \).

Thus, the total favorable outcomes for picking one ball of each color is:
\[ 3 \times 6 \times 7 = 126 \]

### Step 6: Calculate the total outcomes of picking any 3 balls from 16
The total number of ways to choose 3 balls from 16 is given by:
\[ \binom{16}{3} = \frac{16 \times 15 \times 14}{3 \times 2 \times 1} = 560 \]

### Step 7: Calculate the probability
The probability \( P \) of picking one ball of each color is:
\[ P = \frac{\text{Favorable outcomes}}{\text{Total outcomes}} = \frac{126}{560} \]

### Step 8: Simplify the probability
To simplify \( \frac{126}{560} \):
- Both numbers can be divided by 14:
\[ \frac{126 \div 14}{560 \div 14} = \frac{9}{40} \]

### Final Answer
Thus, the probability of getting one ball of each color when three balls are picked at random is:
\[ \frac{9}{40} \]