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 into three groups.

Options

A. A) 34560
B. B) 34650
C. C) 7575
D. D) 5775

Answer

To solve the problem of dividing 12 distinct CDs into 3 equal groups, we can follow these steps:

### Step 1: Understand the Problem
We have 12 distinct CDs that need to be divided into 3 groups, with each group containing 4 CDs.

### Step 2: Calculate the Number of Ways to Choose the First Group
To find the number of ways to select the first group of 4 CDs from the 12, we can use the combination formula:
\[ \text{Number of ways to choose 4 from 12} = \binom{12}{4} = \frac{12!}{4!(12-4)!} = \frac{12!}{4! \cdot 8!} \]

### Step 3: Calculate the Number of Ways to Choose the Second Group
After selecting the first group, we have 8 CDs left. Now, we need to select the second group of 4 CDs from these remaining 8:
\[ \text{Number of ways to choose 4 from 8} = \binom{8}{4} = \frac{8!}{4!(8-4)!} = \frac{8!}{4! \cdot 4!} \]

### Step 4: Calculate the Number of Ways to Choose the Last Group
The last group will automatically consist of the remaining 4 CDs. The number of ways to choose 4 from 4 is:
\[ \text{Number of ways to choose 4 from 4} = \binom{4}{4} = 1 \]

### Step 5: Combine the Results
Now, we multiply the number of ways to choose each group:
\[ \text{Total ways} = \binom{12}{4} \times \binom{8}{4} \times \binom{4}{4} \]

### Step 6: Account for Indistinguishable 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 \(3!\):
\[ \text{Final Total Ways} = \frac{\binom{12}{4} \times \binom{8}{4} \times \binom{4}{4}}{3!} \]

### Step 7: Calculate Each Combination
Now we can calculate each combination:
1. \(\binom{12}{4} = \frac{12!}{4! \cdot 8!} = 495\)
2. \(\binom{8}{4} = \frac{8!}{4! \cdot 4!} = 70\)
3. \(\binom{4}{4} = 1\)

### Step 8: Substitute Back into the Formula
Now substituting these values back:
\[ \text{Total ways} = \frac{495 \times 70 \times 1}{6} = \frac{34650}{6} = 5775 \]

### Conclusion
Thus, the number of ways to divide 12 distinct CDs into 3 equal groups of 4 is **5775**.

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