The number of ways in which 9 persons can be divided into three equal groups is

Options

A. 1680
B. 840
C. 560
D. 280

Answer

To find the number of ways to divide 9 persons into three equal groups, we can follow these steps:

### Step-by-Step Solution:

1. **Choose the first group of 3 persons from 9**:
We can select 3 persons out of 9. This can be calculated using the combination formula \( \binom{n}{r} \), where \( n \) is the total number of persons and \( r \) is the number of persons to choose.
\[ \text{Ways to choose the first group} = \binom{9}{3} \]

2. **Choose the second group of 3 persons from the remaining 6**:
After selecting the first group, we have 6 persons left. We can choose another group of 3 persons from these 6.
\[ \text{Ways to choose the second group} = \binom{6}{3} \]

3. **Choose the third group of 3 persons from the remaining 3**:
After selecting the first two groups, there are 3 persons left. We can choose all of them to form the third group.
\[ \text{Ways to choose the third group} = \binom{3}{3} \]

4. **Calculate the total combinations**:
The total number of ways to form the groups is the product of the combinations calculated in the previous steps:
\[ \text{Total ways} = \binom{9}{3} \times \binom{6}{3} \times \binom{3}{3} \]

5. **Adjust for indistinguishable groups**:
Since the groups are indistinguishable (the order of the groups does not matter), we need to divide by the number of ways to arrange the 3 groups, which is \( 3! \):
\[ \text{Final count} = \frac{\binom{9}{3} \times \binom{6}{3} \times \binom{3}{3}}{3!} \]

6. **Calculate the values**:
- \( \binom{9}{3} = \frac{9!}{3!(9-3)!} = \frac{9 \times 8 \times 7}{3 \times 2 \times 1} = 84 \)
- \( \binom{6}{3} = \frac{6!}{3!(6-3)!} = \frac{6 \times 5 \times 4}{3 \times 2 \times 1} = 20 \)
- \( \binom{3}{3} = 1 \)
- Therefore, total ways = \( 84 \times 20 \times 1 = 1680 \)
- Now divide by \( 3! = 6 \):
\[ \text{Final count} = \frac{1680}{6} = 280 \]

### Final Answer:
The number of ways in which 9 persons can be divided into three equal groups is **280**.