In how many ways can 4 red, 3 yellow and 2 green discs be arranged in a row if the discs of the same colour are indistinguishable?

Options

A. `(9!)/(4!3!2!) `
B. `(10!)/(4!3!2!) `
C. `(9!) `
D. None of these

Explanation

To solve the problem of arranging 4 red, 3 yellow, and 2 green discs in a row, where discs of the same color are indistinguishable, we can follow these steps:

### Step-by-Step Solution:

1. **Count the Total Number of Discs**:
We have:
- 4 red discs
- 3 yellow discs
- 2 green discs

Total number of discs = 4 + 3 + 2 = 9.

2. **Use the Formula for Arrangements**:
The formula for arranging \( n \) items where there are groups of indistinguishable items is given by:
\[ \frac{n!}{n_1! \times n_2! \times n_3!} \]
where \( n \) is the total number of items and \( n_1, n_2, n_3 \) are the counts of indistinguishable items.

In our case:
- \( n = 9 \) (total discs)
- \( n_1 = 4 \) (red discs)
- \( n_2 = 3 \) (yellow discs)
- \( n_3 = 2 \) (green discs)

Thus, the formula becomes:
\[ \frac{9!}{4! \times 3! \times 2!} \]

3. **Calculate Factorials**:
Now we need to calculate the factorials:
- \( 9! = 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 \)
- \( 4! = 4 \times 3 \times 2 \times 1 = 24 \)
- \( 3! = 3 \times 2 \times 1 = 6 \)
- \( 2! = 2 \times 1 = 2 \)

4. **Substitute and Simplify**:
Substitute the factorial values into the formula:
\[ \frac{9!}{4! \times 3! \times 2!} = \frac{362880}{24 \times 6 \times 2} \]
Calculate the denominator:
\[ 24 \times 6 = 144 \]
\[ 144 \times 2 = 288 \]
Now substitute back:
\[ \frac{362880}{288} \]

5. **Perform the Division**:
Now, divide \( 362880 \) by \( 288 \):
\[ 362880 \div 288 = 1260 \]

### Final Answer:
The total number of ways to arrange the discs is **1260**.

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