Two series of a question booklets for an aptitude test are to be given to twelve students. In how many ways can the students be placed in two rows of six each so that there should be no identical series side by side and that the students sitting one behind the other should have the same series?

Options

A. `2 xx ""^(12)C_(6) xx (6 !)^2`
B. `6 ! xx 6!`
C. `7! xx 7!`
D. None of these

Answer

To solve the problem of arranging 12 students in two rows of six each with the given conditions, we can follow these steps:

### Step 1: Understand the Arrangement
We have 12 students and we need to arrange them in two rows of 6 each. Each row will have students with identical series of question booklets. This means that if one student in the front row has a series A booklet, the student directly behind them in the second row must also have a series A booklet.

### Step 2: Define the Series
Let’s denote the two series of booklets as Series 1 and Series 2. Each series will be given to 6 students.

### Step 3: Arranging the Students
1. **Choose 6 students for Series 1**: We can select 6 students out of 12 for Series 1. The number of ways to choose 6 students from 12 is given by the combination formula:
\[ \binom{12}{6} \]

2. **Arrange the Students in Each Series**: Once we have selected 6 students for Series 1, the remaining 6 students will automatically belong to Series 2. We can arrange the 6 students in Series 1 in \(6!\) (factorial of 6) ways, and the same goes for the 6 students in Series 2. Therefore, the arrangements for both series will be:
\[ 6! \text{ (for Series 1)} \times 6! \text{ (for Series 2)} \]

### Step 4: Calculate the Total Arrangements
Putting it all together, the total number of arrangements can be calculated as:
\[ \text{Total arrangements} = \binom{12}{6} \times 6! \times 6! \]

### Step 5: Simplify the Expression
We know that:
\[ \binom{12}{6} = \frac{12!}{6!6!} \]
Thus, the total arrangements can be rewritten as:
\[ \text{Total arrangements} = \frac{12!}{6!6!} \times 6! \times 6! = 12! \]

### Conclusion
The total number of ways to arrange the students under the given conditions is \(6! \times 6!\).

### Final Answer
The correct answer is \(6! \times 6!\).

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