There are 5 red , 4 white and 3 blue marbles in a bag. They are drawn one by one and arranged in a row . Assuming that all the 12 marbles are drawn , determine the number of different arrangements.

Answer

To solve the problem of determining the number of different arrangements of 5 red, 4 white, and 3 blue marbles, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Total Number of Marbles**:
- We have 5 red marbles, 4 white marbles, and 3 blue marbles.
- Total number of marbles = 5 (red) + 4 (white) + 3 (blue) = 12 marbles.

2. **Use the Formula for Arrangements of Indistinguishable Objects**:
- The formula for the number of arrangements of n objects where there are groups of indistinguishable objects is given by:
\[ \text{Number of arrangements} = \frac{n!}{n_1! \cdot n_2! \cdot n_3!} \]
- Here, \( n \) is the total number of marbles, \( n_1 \) is the number of red marbles, \( n_2 \) is the number of white marbles, and \( n_3 \) is the number of blue marbles.
- So, we have:
\[ n = 12, \quad n_1 = 5, \quad n_2 = 4, \quad n_3 = 3 \]

3. **Substituting the Values into the Formula**:
- Substitute the values into the formula:
\[ \text{Number of arrangements} = \frac{12!}{5! \cdot 4! \cdot 3!} \]

4. **Calculate Factorials**:
- Calculate the factorials:
- \( 12! = 479001600 \)
- \( 5! = 120 \)
- \( 4! = 24 \)
- \( 3! = 6 \)

5. **Substituting Factorials into the Equation**:
- Now substitute the factorial values:
\[ \text{Number of arrangements} = \frac{479001600}{120 \cdot 24 \cdot 6} \]

6. **Calculate the Denominator**:
- Calculate the denominator:
\[ 120 \cdot 24 = 2880 \]
\[ 2880 \cdot 6 = 17280 \]

7. **Final Calculation**:
- Now divide the total arrangements by the denominator:
\[ \text{Number of arrangements} = \frac{479001600}{17280} = 27720 \]

### Final Answer:
The number of different arrangements of the marbles is **27720**.