4 gentlemen and 4 ladies take seats at random round a table. The probability that they are sitting alternately is

Options

A. `4//35`
B. `1//70`
C. `2//35`
D. `1//35`

Answer

To solve the problem of finding the probability that 4 gentlemen and 4 ladies sit alternately at a round table, we can follow these steps:

### Step 1: Calculate the total number of arrangements
When arranging \( n \) people around a circular table, the total number of arrangements is given by \( (n-1)! \). Here, we have 8 people (4 gentlemen and 4 ladies).

\[ \text{Total arrangements} = (8-1)! = 7! = 5040 \]

**Hint:** Remember that in circular arrangements, we fix one person to eliminate identical rotations.

### Step 2: Arrange the gentlemen
We can arrange the 4 gentlemen in the circle first. Since we are considering a circular arrangement, we can fix one gentleman and arrange the remaining 3.

\[ \text{Arrangements of gentlemen} = 3! = 6 \]

**Hint:** Fix one person to simplify the arrangement of the rest.

### Step 3: Arrange the ladies
Once the gentlemen are seated, there are 4 gaps created between them where the ladies can sit. We need to arrange the 4 ladies in these 4 gaps.

\[ \text{Arrangements of ladies} = 4! = 24 \]

**Hint:** Each gap between gentlemen can hold one lady, and all ladies can be arranged among themselves.

### Step 4: Calculate the favorable arrangements
The total number of favorable arrangements where gentlemen and ladies sit alternately is the product of the arrangements of gentlemen and ladies.

\[ \text{Favorable arrangements} = \text{Arrangements of gentlemen} \times \text{Arrangements of ladies} = 6 \times 24 = 144 \]

**Hint:** Multiply the arrangements of gentlemen by the arrangements of ladies to find total favorable cases.

### Step 5: Calculate the probability
The probability that the gentlemen and ladies are sitting alternately is given by the ratio of favorable arrangements to total arrangements.

\[ \text{Probability} = \frac{\text{Favorable arrangements}}{\text{Total arrangements}} = \frac{144}{5040} \]

### Step 6: Simplify the probability
Now, we simplify the fraction:

\[ \frac{144}{5040} = \frac{1}{35} \]

**Hint:** Always simplify fractions to their lowest terms to find the final probability.

### Final Answer:
The probability that the 4 gentlemen and 4 ladies are sitting alternately is \( \frac{1}{35} \).