There are 3 red ,4 green and 5 pink 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

Options

A. a. 22770
B. b. 27720
C. c. 22077
D. d. 27270

Answer

To determine the number of different arrangements of the marbles, we can use the formula for permutations of multiset. The formula is given by:

\[ \text{Number of arrangements} = \frac{n!}{n_1! \times n_2! \times n_3!} \]

where:
- \( n \) is the total number of items (marbles in this case),
- \( n_1, n_2, n_3 \) are the counts of each distinct item.

### Step 1: Identify the total number of marbles
We have:
- 3 red marbles
- 4 green marbles
- 5 pink marbles

Calculating the total number of marbles:
\[ n = 3 + 4 + 5 = 12 \]

### Step 2: Identify the factorials for each color of marbles
Now we need to calculate the factorials for the counts of each color:
- For red marbles: \( n_1 = 3 \) so \( n_1! = 3! = 6 \)
- For green marbles: \( n_2 = 4 \) so \( n_2! = 4! = 24 \)
- For pink marbles: \( n_3 = 5 \) so \( n_3! = 5! = 120 \)

### Step 3: Apply the formula
Now we can substitute these values into the formula:
\[ \text{Number of arrangements} = \frac{12!}{3! \times 4! \times 5!} \]

### Step 4: Calculate \( 12! \)
Calculating \( 12! \):
\[ 12! = 479001600 \]

### Step 5: Calculate the denominator
Now calculate the denominator:
\[ 3! \times 4! \times 5! = 6 \times 24 \times 120 \]
Calculating step by step:
- First, \( 6 \times 24 = 144 \)
- Then, \( 144 \times 120 = 17280 \)

### Step 6: Final calculation
Now we can calculate the number of arrangements:
\[ \text{Number of arrangements} = \frac{479001600}{17280} = 27720 \]

Thus, the total number of different arrangements of the marbles is **27720**.