In how many ways, can 15 people be seated around two round tables with seating capacities of 7 and 8 people?

Options

A. 15!/(8!)
B. 71/88!
C. `""^(15)C_(8)xx6!xx7!`
D. `""^(15)C_(8)**8!`

Answer

To solve the problem of seating 15 people around two round tables with seating capacities of 7 and 8 people, we can follow these steps:

### Step 1: Choose the Groups
First, we need to select which 7 people will sit at the first table (the table with a capacity of 7). The remaining 8 people will automatically go to the second table.

The number of ways to choose 7 people from 15 is given by the combination formula:

\[ \text{Number of ways to choose 7 from 15} = \binom{15}{7} \]

### Step 2: Arrange the People at the Tables
Next, we need to arrange the selected groups around their respective tables.

- For the table with 7 people, since it is a round table, the number of arrangements is given by \((n-1)!\), where \(n\) is the number of people. Therefore, the number of ways to arrange 7 people around a round table is:

\[ (7-1)! = 6! \]

- For the table with 8 people, similarly, the number of arrangements is:

\[ (8-1)! = 7! \]

### Step 3: Combine the Results
Now, we combine the results from the previous steps to find the total number of ways to seat the 15 people:

\[ \text{Total arrangements} = \binom{15}{7} \times 6! \times 7! \]

### Step 4: Calculate the Values
Now we can compute the values:

1. Calculate \(\binom{15}{7}\):
\[ \binom{15}{7} = \frac{15!}{7!(15-7)!} = \frac{15!}{7! \times 8!} \]

2. Calculate \(6!\) and \(7!\):
\[ 6! = 720 \]
\[ 7! = 5040 \]

3. Substitute these values into the total arrangements formula:
\[ \text{Total arrangements} = \binom{15}{7} \times 720 \times 5040 \]

### Final Calculation
Now, we can calculate the final answer using the values we have:

1. Find \(\binom{15}{7}\):
\[ \binom{15}{7} = 6435 \]

2. Now, substitute:
\[ \text{Total arrangements} = 6435 \times 720 \times 5040 \]

3. Calculate the final product:
\[ \text{Total arrangements} = 6435 \times 720 \times 5040 = 2,329,920,000 \]

Thus, the total number of ways to seat 15 people around two round tables with seating capacities of 7 and 8 is **2,329,920,000**.

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