Study the given information carefully to answer the question that follow. An urn contains 4 green, 5 blue, 2 red and 3 yellow marbles.
If eight marbles are drawn at random, what is the probability that there are equal number of marbles of each colour ?

Options

A. A) `(4)/(7)`
B. B) `(361)/(728)`
C. C) `(60)/(1001)`
D. D) `(1)/(1)`

Answer

To solve the problem, we need to find the probability of drawing 8 marbles from an urn containing 4 green, 5 blue, 2 red, and 3 yellow marbles such that there are equal numbers of each color.

### Step-by-step Solution:

1. **Identify the total number of marbles:**
- Green: 4
- Blue: 5
- Red: 2
- Yellow: 3
- Total = 4 + 5 + 2 + 3 = 14 marbles.

2. **Determine the required condition:**
- We need to draw 8 marbles such that there are equal numbers of each color. Since there are 4 colors, we need 2 marbles of each color (2 Green, 2 Blue, 2 Red, and 2 Yellow).

3. **Calculate the total number of ways to choose 8 marbles from 14:**
- The total ways to choose 8 marbles from 14 can be calculated using the combination formula \( \binom{n}{r} \):
\[ \text{Total ways} = \binom{14}{8} = \binom{14}{6} = \frac{14!}{6! \cdot 8!} \]
- This simplifies to:
\[ \binom{14}{6} = \frac{14 \times 13 \times 12 \times 11 \times 10 \times 9}{6 \times 5 \times 4 \times 3 \times 2 \times 1} = 3003 \]

4. **Calculate the number of favorable outcomes:**
- We need to find the number of ways to select 2 marbles of each color:
- For Green: \( \binom{4}{2} \)
- For Blue: \( \binom{5}{2} \)
- For Red: \( \binom{2}{2} \)
- For Yellow: \( \binom{3}{2} \)
- Calculate each:
- \( \binom{4}{2} = 6 \)
- \( \binom{5}{2} = 10 \)
- \( \binom{2}{2} = 1 \)
- \( \binom{3}{2} = 3 \)
- Multiply these together to find the total favorable outcomes:
\[ \text{Favorable outcomes} = 6 \times 10 \times 1 \times 3 = 180 \]

5. **Calculate the probability:**
- The probability \( P \) of drawing 8 marbles with 2 of each color is given by:
\[ P = \frac{\text{Favorable outcomes}}{\text{Total ways}} = \frac{180}{3003} \]
- Simplifying this fraction:
\[ P = \frac{60}{1001} \]

### Final Answer:
The probability that there are equal numbers of marbles of each color when drawing 8 marbles is \( \frac{60}{1001} \).