The number of ways in which 6 hindus and 6 muslim sits around a round table so that two hindus can never sit together is

Options

A. 5!.6!
B. 5!.5!
C. 6!.6!
D. none of these

Answer

To find the number of ways in which 6 Hindus and 6 Muslims can sit around a round table such that no two Hindus sit together, we can follow these steps:

### Step 1: Fix one Hindu
Since the arrangement is circular, we can fix one Hindu in one position to eliminate the effect of rotations. This means we have 5 remaining Hindus to arrange.

**Hint:** Fixing one person in a circular arrangement helps to simplify the counting by removing identical rotations.

### Step 2: Arrange the remaining Hindus
After fixing one Hindu, we have 5 Hindus left to arrange. The number of ways to arrange these 5 Hindus is given by \(5!\).

**Hint:** The factorial notation \(n!\) represents the number of ways to arrange \(n\) distinct objects.

### Step 3: Place the Muslims
With one Hindu fixed, there will be 6 gaps created around the table (one gap between each pair of seated Hindus). Since we want to ensure that no two Hindus sit together, we can place the 6 Muslims in these 6 gaps.

The number of ways to arrange the 6 Muslims is given by \(6!\).

**Hint:** Gaps created by fixed positions allow for the arrangement of other groups without violating the seating condition.

### Step 4: Calculate the total arrangements
The total number of arrangements is the product of the arrangements of the Hindus and the Muslims. Therefore, the total number of ways is:

\[ 5! \times 6! \]

### Step 5: Compute the factorials
Now we can compute the values:
- \(5! = 120\)
- \(6! = 720\)

So, the total arrangements will be:

\[ 5! \times 6! = 120 \times 720 = 86400 \]

### Final Answer
Thus, the number of ways in which 6 Hindus and 6 Muslims can sit around a round table such that no two Hindus sit together is **86400**.

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