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 into two groups of 6 CDs, 4 CDs and 2CDs.

Options

A. 12680
B. 18360
C. 13860
D. `(12!)/(6!+4!+2!)`

Answer

To solve the problem of dividing 12 distinct CDs into groups of 6, 4, and 2, we can follow these steps:

### Step 1: Choose 6 CDs from 12
We start by selecting 6 CDs from the total of 12. The number of ways to choose 6 CDs from 12 is given by the combination formula \( \binom{n}{r} \), which is defined as:

\[ \binom{n}{r} = \frac{n!}{r!(n-r)!} \]

For our case, we have:

\[ \binom{12}{6} = \frac{12!}{6! \cdot (12-6)!} = \frac{12!}{6! \cdot 6!} \]

### Step 2: Choose 4 CDs from the remaining 6
After selecting the first group of 6 CDs, we now have 6 CDs left. We need to choose 4 from these 6. The number of ways to do this is:

\[ \binom{6}{4} = \frac{6!}{4! \cdot (6-4)!} = \frac{6!}{4! \cdot 2!} \]

### Step 3: The remaining CDs
After selecting 4 CDs from the remaining 6, there will be 2 CDs left. Since we need to select all 2 CDs, the number of ways to choose 2 from 2 is:

\[ \binom{2}{2} = 1 \]

### Step 4: Combine the results
Now, we can combine the results from the three steps. The total number of ways to divide the CDs into groups of 6, 4, and 2 is the product of the combinations calculated in the previous steps:

\[ \text{Total ways} = \binom{12}{6} \times \binom{6}{4} \times \binom{2}{2} \]

### Step 5: Calculate the values
Now we calculate each of these combinations:

1. Calculate \( \binom{12}{6} \):
\[ \binom{12}{6} = \frac{12!}{6! \cdot 6!} = \frac{479001600}{720 \cdot 720} = 924 \]

2. Calculate \( \binom{6}{4} \):
\[ \binom{6}{4} = \frac{6!}{4! \cdot 2!} = \frac{720}{24 \cdot 2} = 15 \]

3. Calculate \( \binom{2}{2} \):
\[ \binom{2}{2} = 1 \]

### Step 6: Final Calculation
Now we multiply these values together:

\[ \text{Total ways} = 924 \times 15 \times 1 = 13860 \]

Thus, the total number of ways to divide the 12 distinct CDs into groups of 6, 4, and 2 is **13860**.

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