There are 3 blue balls, 4 red balls and 5 green balls. In how many ways can they be arranged in a row?

Explanation

To solve the problem of arranging 3 blue balls, 4 red balls, and 5 green balls in a row, we will follow these steps:

### Step 1: Calculate the Total Number of Balls
First, we need to find the total number of balls. We have:
- Blue balls = 3
- Red balls = 4
- Green balls = 5

Total number of balls = 3 + 4 + 5 = 12

### Step 2: Use the Formula for Arrangements of Indistinguishable Objects
The formula to find the number of arrangements of n objects where there are groups of indistinguishable objects is given by:

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

Where:
- \( n \) is the total number of objects,
- \( n_1, n_2, n_3 \) are the counts of each indistinguishable group.

In our case:
- \( n = 12 \) (total balls)
- \( n_1 = 3 \) (blue balls)
- \( n_2 = 4 \) (red balls)
- \( n_3 = 5 \) (green balls)

### Step 3: Substitute the Values into the Formula
Substituting the values into the formula gives us:

\[ \text{Number of arrangements} = \frac{12!}{3! \times 4! \times 5!} \]

### Step 4: Calculate Factorials
Now, we need to calculate the factorials:

- \( 12! = 479001600 \)
- \( 3! = 6 \)
- \( 4! = 24 \)
- \( 5! = 120 \)

### Step 5: Calculate the Denominator
Now, calculate the product of the factorials in the denominator:

\[ 3! \times 4! \times 5! = 6 \times 24 \times 120 \]

Calculating this step-by-step:
- \( 6 \times 24 = 144 \)
- \( 144 \times 120 = 17280 \)

### Step 6: Divide the Factorials
Now, we can find the number of arrangements:

\[ \text{Number of arrangements} = \frac{479001600}{17280} = 27720 \]

### Final Answer
Thus, the total number of ways to arrange the balls in a row is **27,720**.

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