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 among three French students equally.

Options

A. 34650
B. 272100
C. 7575
D. 5775

Answer

To solve the problem of distributing 12 distinct CDs among 3 students equally, we can follow these steps:

### Step 1: Determine the number of CDs each student receives
Since there are 12 CDs and 3 students, each student will receive:
\[ \text{Number of CDs per student} = \frac{12}{3} = 4 \]

### Step 2: Calculate the number of ways to distribute the CDs
We need to find the number of ways to choose 4 CDs for the first student from the 12 available CDs. This can be done using the combination formula:
\[ \text{Ways to choose 4 CDs from 12} = \binom{12}{4} \]

### Step 3: Calculate the remaining CDs for the second student
After giving 4 CDs to the first student, we have:
\[ 12 - 4 = 8 \text{ CDs left} \]
Now, we need to choose 4 CDs for the second student from these 8 remaining CDs:
\[ \text{Ways to choose 4 CDs from 8} = \binom{8}{4} \]

### Step 4: Assign the remaining CDs to the third student
After distributing 4 CDs to the first and 4 to the second student, there will be:
\[ 8 - 4 = 4 \text{ CDs left} \]
The third student will receive all of these 4 CDs, and there is only 1 way to do this:
\[ \text{Ways to choose 4 CDs from 4} = \binom{4}{4} = 1 \]

### Step 5: Calculate the total number of distributions
The total number of ways to distribute the CDs is given by multiplying the number of ways for each step:
\[ \text{Total ways} = \binom{12}{4} \times \binom{8}{4} \times \binom{4}{4} \]

### Step 6: Calculate the combinations
Now, we can calculate the combinations:
\[ \binom{12}{4} = \frac{12!}{4!(12-4)!} = \frac{12!}{4! \cdot 8!} = \frac{12 \times 11 \times 10 \times 9}{4 \times 3 \times 2 \times 1} = 495 \]
\[ \binom{8}{4} = \frac{8!}{4!(8-4)!} = \frac{8!}{4! \cdot 4!} = \frac{8 \times 7 \times 6 \times 5}{4 \times 3 \times 2 \times 1} = 70 \]
\[ \binom{4}{4} = 1 \]

### Step 7: Multiply the results
Now, we multiply the results from the combinations:
\[ \text{Total ways} = 495 \times 70 \times 1 = 34650 \]

### Step 8: Adjust for indistinguishable groups
Since the students are indistinguishable in terms of who receives which set of CDs, we need to divide by the number of ways to arrange the 3 students (which is \(3!\)):
\[ \text{Final Total Ways} = \frac{34650}{3!} = \frac{34650}{6} = 5775 \]

### Conclusion
Thus, the total number of ways to distribute the 12 distinct CDs among 3 students equally is:
\[ \boxed{5775} \]