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 two French students who had recently visited FTII from France such that one student has 8 CDs and another one has 4 CDs.

Options

A. 990
B. 900
C. 945
D. 1188

Answer

To solve the problem of distributing 12 distinct CDs between two students such that one student receives 8 CDs and the other receives 4 CDs, we can follow these steps:

### Step 1: Choose CDs for the First Student
We need to select 8 CDs out of the 12 available CDs for the first student. The number of ways to choose 8 CDs from 12 is given by the combination formula:

\[ \text{Number of ways} = \binom{12}{8} \]

### Step 2: Calculate the Combination
Using the combination formula, we can calculate:

\[ \binom{12}{8} = \frac{12!}{8!(12-8)!} = \frac{12!}{8!4!} \]

### Step 3: Choose CDs for the Second Student
Once we have chosen 8 CDs for the first student, the remaining 4 CDs automatically go to the second student. The number of ways to choose these 4 CDs from the remaining 4 is:

\[ \binom{4}{4} = 1 \]

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

\[ \text{Total ways} = \binom{12}{8} \times \binom{4}{4} = \binom{12}{8} \times 1 = \binom{12}{8} \]

### Step 5: Factorial Calculation
Now we will calculate \(\binom{12}{8}\):

\[ \binom{12}{8} = \frac{12 \times 11 \times 10 \times 9}{4 \times 3 \times 2 \times 1} = \frac{11880}{24} = 495 \]

### Step 6: Account for Distinct Students
Since the two students are distinct, we need to multiply the result by \(2!\) (the number of ways to arrange the two students):

\[ \text{Final Total} = 495 \times 2 = 990 \]

### Conclusion
Thus, the total number of ways to distribute the 12 distinct CDs to the two students such that one gets 8 and the other gets 4 is:

\[ \boxed{990} \]

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