In a single occupancy co-ed hostel, consisting of 20 rooms, exactly 10 guys and 10 girls are allowed to stay in. Each one of them stays in a separate room. The hostel is designed in a circular form such that the door of a room opens opposite the door of another room. In how many ways can the rooms be allocated to them so that whenever a guy looks either side of his door there is always a girl next door?

Options

A. A) `2!xx(10!)^2`
B. B) `20!-19!`
C. C) `9! xx 10!`
D. D) `2!xx(19!)/(9!xx10!)`

Answer

To solve the problem of allocating rooms in a circular hostel such that every guy has a girl next to him, we can follow these steps:

### Step 1: Understand the Arrangement
We have a circular arrangement of 20 rooms, with 10 guys and 10 girls. The requirement is that every guy must have a girl next to him on both sides.

### Step 2: Arrange the Girls
Since we need to ensure that there is always a girl next to a guy, we can start by placing the girls in the rooms. We can arrange the 10 girls in the 10 rooms, leaving one room between each girl empty. This means we will occupy every alternate room with a girl.

### Step 3: Calculate the Ways to Arrange Girls
The number of ways to arrange the 10 girls in the 10 selected rooms is given by \(10!\) (10 factorial).

### Step 4: Place the Guys
Once the girls are placed, the remaining 10 rooms (which are the alternate rooms) will be occupied by the guys. Since there are no restrictions on how the guys can be arranged in their rooms, we can arrange the 10 guys in the 10 rooms in \(10!\) ways.

### Step 5: Consider Room Exchange
Since the arrangement is circular, each pair of adjacent rooms (one girl and one guy) can switch their positions. This means for each pair of a girl and a guy, there are 2 ways to arrange them (girl in the first room and guy in the second room, or vice versa).

### Step 6: Calculate Total Arrangements
Since there are 10 pairs (one girl and one guy), the total arrangements considering the exchange is given by:
\[ 2^{10} \text{ (for each pair)} \times 10! \text{ (for girls)} \times 10! \text{ (for guys)} \]

### Final Calculation
Thus, the total number of ways to allocate the rooms is:
\[ 2^{10} \times (10!)^2 \]

### Conclusion
The final answer is:
\[ 2^{10} \times (10!)^2 \]

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