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 distributed to two French students equally.

Options

A. 594
B. 990
C. 984
D. 924

Answer

To solve the problem of distributing 12 distinct CDs equally between two students, we can follow these steps:

### Step 1: Determine the number of CDs each student receives
Since we have 12 distinct CDs and we want to distribute them equally between two students, each student will receive 6 CDs.

### Step 2: Calculate the number of ways to choose 6 CDs from 12
We can use the combination formula to find the number of ways to choose 6 CDs from 12. The formula for combinations is given by:

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

In our case, \( n = 12 \) and \( r = 6 \):

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

### Step 3: Calculate \( \binom{12}{6} \)
Now we can calculate \( \binom{12}{6} \):

\[ \binom{12}{6} = \frac{12 \times 11 \times 10 \times 9 \times 8 \times 7}{6 \times 5 \times 4 \times 3 \times 2 \times 1} = \frac{665280}{720} = 924 \]

### Step 4: Account for indistinguishable groups
Since the two groups of CDs (one for each student) are indistinguishable, we need to divide the result by 2 to avoid counting the same distribution twice. This gives us:

\[ \text{Total ways} = \frac{924}{2} = 462 \]

### Final Answer
Thus, the number of ways to distribute the 12 distinct CDs equally between two students is **462**.

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