Number of ways in which 12 different balls can be divided into groups of 5, 4 and 3 balls are

Options

A. `(12!)/(5!4!)`
B. `(12 !)/(5! 4! 3!)`
C. `(12!)/(5! 4! 3!3!)`
D. None of these

Answer

To solve the problem of dividing 12 different balls into groups of 5, 4, and 3, we can follow these steps:

### Step 1: Understand the Problem
We have 12 different balls, and we want to divide them into three groups: one group of 5 balls, another group of 4 balls, and a third group of 3 balls.

### Step 2: Use the Formula for Permutations
The number of ways to divide n different items into groups of sizes \( n_1, n_2, n_3 \) is given by the formula:

\[ \frac{n!}{n_1! \times n_2! \times n_3!} \]

In our case, \( n = 12 \), \( n_1 = 5 \), \( n_2 = 4 \), and \( n_3 = 3 \).

### Step 3: Substitute the Values into the Formula
Substituting the values into the formula, we get:

\[ \frac{12!}{5! \times 4! \times 3!} \]

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

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

### Step 5: Substitute the Factorials Back into the Equation
Now we substitute these values back into our equation:

\[ \frac{479001600}{120 \times 24 \times 6} \]

### Step 6: Calculate the Denominator
Calculating the denominator:

\[ 120 \times 24 = 2880 \]
\[ 2880 \times 6 = 17280 \]

### Step 7: Divide the Numerator by the Denominator
Now we divide the numerator by the denominator:

\[ \frac{479001600}{17280} = 27720 \]

### Final Answer
Thus, the number of ways to divide 12 different balls into groups of 5, 4, and 3 is **27720**.

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