In how many ways, can 24 persons be seated around a circular table, if there are 13 seats?

Options

A. a) `(24!)/(13xx11!)`
B. b) `(22!)/(14xx12!)`
C. c)`(23!)/(13xx11!)`
D. `d )(24!)/(12xx12!)`

Answer

To solve the problem of how many ways 24 persons can be seated around a circular table with 13 seats, we can follow these steps:

### Step 1: Choose 13 persons from 24
Since there are 24 persons and only 13 seats available, we first need to select which 13 persons will be seated. This can be done using combinations.

The number of ways to choose 13 persons from 24 is given by the combination formula:
\[ \binom{n}{r} = \frac{n!}{r!(n-r)!} \]
Here, \( n = 24 \) and \( r = 13 \):
\[ \binom{24}{13} = \frac{24!}{13!(24-13)!} = \frac{24!}{13! \cdot 11!} \]

### Step 2: Arrange the 13 persons around the circular table
When arranging persons in a circle, we fix one person to eliminate the effect of rotations. The number of ways to arrange \( r \) persons in a circle is given by \( (r-1)! \).

For our case with 13 persons:
\[ \text{Ways to arrange 13 persons in a circle} = (13 - 1)! = 12! \]

### Step 3: Combine the two results
Now, we multiply the number of ways to choose the persons by the number of ways to arrange them:
\[ \text{Total arrangements} = \binom{24}{13} \times 12! \]

### Step 4: Substitute the values
Substituting the combination formula we derived earlier:
\[ \text{Total arrangements} = \frac{24!}{13! \cdot 11!} \times 12! \]

### Step 5: Simplify the expression
We can simplify the expression:
\[ \text{Total arrangements} = \frac{24! \cdot 12!}{13! \cdot 11!} \]

This is the final expression representing the total number of ways to seat 24 persons in 13 seats around a circular table.

### Final Answer
Thus, the total number of ways to seat 24 persons around a circular table with 13 seats is:
\[ \frac{24! \cdot 12!}{13! \cdot 11!} \]

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