Find the number of ways in which 12 students may be equally divided into three groups .

Options

A. 3355
B. 5775
C. 7755
D. 8822

Answer

To find the number of ways to divide 12 students into three equal groups of 4 students each, we can follow these steps:

### Step 1: Determine the total number of students and groups
We have a total of 12 students and we want to divide them into 3 groups. Each group will contain 4 students.

### Step 2: Calculate the number of ways to choose the first group
To choose the first group of 4 students from 12, we can use the combination formula:
\[ \binom{n}{r} = \frac{n!}{r!(n-r)!} \]
For the first group:
\[ \binom{12}{4} = \frac{12!}{4!(12-4)!} = \frac{12!}{4!8!} \]

### Step 3: Calculate the number of ways to choose the second group
After selecting the first group, we have 8 students left. We now choose the second group of 4 students from these 8:
\[ \binom{8}{4} = \frac{8!}{4!(8-4)!} = \frac{8!}{4!4!} \]

### Step 4: The remaining students form the third group
After selecting the first two groups, the remaining 4 students will automatically form the third group. There is only 1 way to choose this group.

### Step 5: Account for the indistinguishability of the groups
Since the groups are indistinguishable (i.e., the order of the groups does not matter), we need to divide by the number of ways to arrange the 3 groups, which is \(3!\):
\[ 3! = 6 \]

### Step 6: Combine all calculations
Now we can combine all these calculations:
\[ \text{Total ways} = \frac{\binom{12}{4} \times \binom{8}{4}}{3!} \]

Calculating each part:
1. Calculate \(\binom{12}{4}\):
\[ \binom{12}{4} = \frac{12 \times 11 \times 10 \times 9}{4 \times 3 \times 2 \times 1} = 495 \]

2. Calculate \(\binom{8}{4}\):
\[ \binom{8}{4} = \frac{8 \times 7 \times 6 \times 5}{4 \times 3 \times 2 \times 1} = 70 \]

3. Now plug these values into the total ways formula:
\[ \text{Total ways} = \frac{495 \times 70}{6} \]

4. Calculate \(495 \times 70\):
\[ 495 \times 70 = 34650 \]

5. Finally, divide by 6:
\[ \frac{34650}{6} = 5775 \]

### Final Answer
The number of ways to divide 12 students into three groups of 4 is **5775**.

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