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 such that one group has 8 CDs and another one has 4 CDs.

Options

A. 594
B. 495
C. 945
D. 32

Answer

To solve the problem of dividing 12 distinct CDs into two groups, one with 8 CDs and the other with 4 CDs, we can use the concept of combinations. Here’s the step-by-step solution:

### Step 1: Understand the Problem
We have 12 distinct CDs, and we need to select 8 CDs for the first group. The remaining 4 CDs will automatically form the second group.

### Step 2: Use the Combination Formula
The number of ways to choose \( r \) items from \( n \) distinct items is given by the combination formula:

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

In our case, we need to find \( C(12, 8) \).

### Step 3: Calculate \( C(12, 8) \)
Using the combination formula:

\[ C(12, 8) = \frac{12!}{8!(12 - 8)!} = \frac{12!}{8! \cdot 4!} \]

### Step 4: Simplify the Factorials
We can simplify \( 12! \):

\[ 12! = 12 \times 11 \times 10 \times 9 \times 8! \]

So, substituting this back into our combination formula:

\[ C(12, 8) = \frac{12 \times 11 \times 10 \times 9 \times 8!}{8! \cdot 4!} \]

### Step 5: Cancel Out \( 8! \)
Now, we can cancel \( 8! \) from the numerator and denominator:

\[ C(12, 8) = \frac{12 \times 11 \times 10 \times 9}{4!} \]

### Step 6: Calculate \( 4! \)
Now, we calculate \( 4! \):

\[ 4! = 4 \times 3 \times 2 \times 1 = 24 \]

### Step 7: Substitute \( 4! \) Back
Now we can substitute \( 4! \) back into our equation:

\[ C(12, 8) = \frac{12 \times 11 \times 10 \times 9}{24} \]

### Step 8: Perform the Multiplication
Calculating the numerator:

\[ 12 \times 11 = 132 \]
\[ 132 \times 10 = 1320 \]
\[ 1320 \times 9 = 11880 \]

### Step 9: Divide by 24
Now we divide by 24:

\[ C(12, 8) = \frac{11880}{24} = 495 \]

### Final Answer
Thus, the number of ways to divide the 12 distinct CDs into two groups of 8 and 4 is **495**.

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