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 gifted to three French students who had recently visited FTII from France such that one student has 6 CDs, second student has 4 CDs and the third one has 2 CDs.

Options

A. 31860
B. 82160
C. 13860
D. 83160

Answer

To solve the problem of distributing 12 distinct CDs among three students such that one student receives 6 CDs, another receives 4 CDs, and the last one receives 2 CDs, we can follow these steps:

### Step 1: Choose CDs for the First Student
We start by selecting 6 CDs out of the 12 for the first student. The number of ways to choose 6 CDs from 12 is given by the combination formula:

\[ \text{Ways for first student} = \binom{12}{6} \]

### Step 2: Choose CDs for the Second Student
After assigning 6 CDs to the first student, we have 6 CDs left. Now, we need to choose 4 CDs for the second student. The number of ways to choose 4 CDs from the remaining 6 is:

\[ \text{Ways for second student} = \binom{6}{4} \]

### Step 3: Assign Remaining CDs to the Third Student
After distributing CDs to the first and second students, we will have 2 CDs left, which will automatically go to the third student. The number of ways to choose 2 CDs from 2 is:

\[ \text{Ways for third student} = \binom{2}{2} \]

### Step 4: Calculate Total Ways
Now, we multiply the number of ways for each student to find the total number of ways to distribute the CDs:

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

### Step 5: Consider the Arrangement of Students
Since the students are distinct, we also need to account for the different arrangements of the students. There are 3 students, so we multiply by the number of ways to arrange 3 students, which is \(3!\):

\[ \text{Total Arrangements} = 3! \]

### Step 6: Final Calculation
Putting it all together, the final formula for the total number of ways to distribute the CDs is:

\[ \text{Final Total Ways} = \left( \binom{12}{6} \times \binom{6}{4} \times \binom{2}{2} \right) \times 3! \]

Now, we can calculate each component:

1. \(\binom{12}{6} = \frac{12!}{6! \times 6!} = 924\)
2. \(\binom{6}{4} = \frac{6!}{4! \times 2!} = 15\)
3. \(\binom{2}{2} = 1\)
4. \(3! = 6\)

Now substituting these values into the total ways calculation:

\[ \text{Total Ways} = 924 \times 15 \times 1 \times 6 = 83,160 \]

Thus, the total number of ways to distribute the CDs is **83,160**.