Answers these questions based on the following information.
A film library at FTII (Film and Television Institute of India) Pune has 12 distinct CDs on French cinema.
Find the number of ways in which these CDs can be divided equally equally into two groups.

Options

A. 1024
B. 492
C. 462
D. 924

Answer

To solve the problem of dividing 12 distinct CDs into two equal groups, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Problem**: We need to divide 12 distinct CDs into two groups of 6 each.

2. **Choosing the First Group**: We can choose 6 CDs out of 12 to form the first group. The number of ways to choose 6 CDs from 12 can be calculated using the combination formula:
\[ \binom{n}{r} = \frac{n!}{r!(n-r)!} \]
Here, \( n = 12 \) and \( r = 6 \):
\[ \binom{12}{6} = \frac{12!}{6! \cdot 6!} \]

3. **Calculating the Combination**:
- First, calculate \( 12! \):
\[ 12! = 12 \times 11 \times 10 \times 9 \times 8 \times 7 \times 6! \]
- Substitute this into the combination formula:
\[ \binom{12}{6} = \frac{12 \times 11 \times 10 \times 9 \times 8 \times 7 \times 6!}{6! \cdot 6!} = \frac{12 \times 11 \times 10 \times 9 \times 8 \times 7}{6!} \]
- Calculate \( 6! \):
\[ 6! = 720 \]
- Now, calculate \( \binom{12}{6} \):
\[ \binom{12}{6} = \frac{665280}{720} = 924 \]

4. **Accounting for Group Order**: Since the two groups are indistinguishable (i.e., Group 1 and Group 2 are not labeled), we need to divide by 2 to avoid counting the same division twice:
\[ \text{Total Ways} = \frac{924}{2} = 462 \]

### Final Answer:
The number of ways to divide the 12 distinct CDs into two equal groups of 6 is **462**.

---