In how many ways can 20 persons be seated round a table if there are 9 chairs.

Answer

To solve the problem of how many ways 20 persons can be seated around a table with 9 chairs, we can break down the solution into clear steps:

### Step-by-Step Solution:

1. **Identify the Problem**: We have 20 persons and 9 chairs. We need to find out how many ways we can select and arrange 9 persons from the 20 around a round table.

2. **Select 9 Persons from 20**: The first step is to choose 9 persons out of the 20. This can be done using the combination formula:
\[ \text{Number of ways to choose 9 persons from 20} = \binom{20}{9} \]

3. **Arrange 9 Persons Around a Round Table**: Once we have selected 9 persons, we need to arrange them around a round table. The number of ways to arrange \( n \) persons around a round table is given by \( (n-1)! \). In our case, since we have 9 persons, the number of arrangements is:
\[ \text{Number of arrangements} = (9-1)! = 8! \]

4. **Combine the Two Steps**: Now, we combine the number of ways to choose the persons and the number of ways to arrange them:
\[ \text{Total ways} = \binom{20}{9} \times 8! \]

5. **Calculate the Values**:
- First, calculate \( \binom{20}{9} \):
\[ \binom{20}{9} = \frac{20!}{9!(20-9)!} = \frac{20!}{9! \times 11!} \]
- Next, calculate \( 8! \):
\[ 8! = 40320 \]

6. **Final Calculation**: Now, multiply the two results:
\[ \text{Total ways} = \binom{20}{9} \times 8! \]

### Final Answer:
The total number of ways to seat 20 persons in 9 chairs around a table is:
\[ \text{Total ways} = \binom{20}{9} \times 8! \]