In how many ways can 7 Englishmen and 7 Americans sit down at a round table, no 2 Americans being in consecutive positions?

Options

A. 3628800
B. 2628800
C. 3628000
D. 3328800

Answer

To solve the problem of how many ways 7 Englishmen and 7 Americans can sit down at a round table with the condition that no two Americans are sitting next to each other, we can follow these steps:

### Step 1: Arrange the Englishmen
Since we are arranging people around a round table, we can fix one Englishman to eliminate the effect of rotations. This means we only need to arrange the remaining 6 Englishmen.

- The number of ways to arrange 6 Englishmen is given by \(6!\).

### Step 2: Calculate \(6!\)
Calculating \(6!\):

\[ 6! = 720 \]

### Step 3: Create Spaces for Americans
Once the Englishmen are arranged, they create spaces for the Americans. With 7 Englishmen seated, there will be 7 gaps between them (including the gap before the first Englishman and after the last one).

### Step 4: Place the Americans
We need to place 7 Americans in these 7 gaps, ensuring that no two Americans sit next to each other. Since there are exactly 7 gaps and 7 Americans, each gap will contain exactly one American.

- The number of ways to arrange the 7 Americans in the 7 gaps is given by \(7!\).

### Step 5: Calculate \(7!\)
Calculating \(7!\):

\[ 7! = 5040 \]

### Step 6: Total Arrangements
Now, we can find the total arrangements by multiplying the arrangements of the Englishmen and the Americans:

\[ \text{Total arrangements} = 6! \times 7! = 720 \times 5040 \]

### Step 7: Calculate the Total
Calculating the total:

\[ 720 \times 5040 = 3628800 \]

### Final Answer
Thus, the total number of ways in which 7 Englishmen and 7 Americans can sit at a round table, with no two Americans sitting next to each other, is:

\[ \boxed{3628800} \]

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