The number of ways in which 20 girls be seated round a table if there are only 10 chairs

Options

A. `(""^(20)C_(10))/(2)`
B. `""^(20)C_(10)9!`
C. `(""^(20)C_(10))/(10)`
D. `(""^(20)P_(10))/(10)`

Answer

To solve the problem of seating 20 girls around a round table with only 10 chairs, we can follow these steps:

### Step-by-Step Solution:

1. **Select 10 Girls from 20:**
We first need to choose 10 girls out of the 20 available. The number of ways to choose 10 girls from 20 is given by the combination formula:
\[ \binom{20}{10} \]

2. **Arrange the Selected Girls:**
After selecting the 10 girls, we need to arrange them in the 10 chairs around the round table. The arrangement of \( n \) objects in a circle is given by \( (n-1)! \). Therefore, for 10 girls, the number of arrangements is:
\[ (10 - 1)! = 9! \]

3. **Total Ways to Seat the Girls:**
The total number of ways to seat the girls is the product of the number of ways to choose the girls and the number of ways to arrange them:
\[ \text{Total Ways} = \binom{20}{10} \times 9! \]

4. **Calculating the Combination:**
We can express the combination \( \binom{20}{10} \) using factorials:
\[ \binom{20}{10} = \frac{20!}{10! \times 10!} \]

5. **Final Expression:**
Substituting this back into our total ways expression, we get:
\[ \text{Total Ways} = \frac{20!}{10! \times 10!} \times 9! \]

6. **Simplifying the Expression:**
We can simplify this further:
\[ \text{Total Ways} = \frac{20! \times 9!}{10! \times 10!} \]

7. **Understanding the Result:**
This expression can also be interpreted as:
\[ \text{Total Ways} = \frac{20P10}{10} \]
where \( 20P10 \) is the number of permutations of 20 items taken 10 at a time.

### Final Answer:
Thus, the total number of ways to seat 20 girls in 10 chairs around a round table is:
\[ \text{Total Ways} = \binom{20}{10} \times 9! \]