In a group of 8 girls, two girls are sisters. The number of ways in which the girls can sit in a row so that two sisters are not sitting together is:

Options

A. 4820
B. 1410
C. 2830
D. 30240

Answer

To solve the problem of how many ways 8 girls can sit in a row such that two sisters are not sitting together, we can follow these steps:

### Step 1: Calculate the total arrangements of the 8 girls
The total number of ways to arrange 8 girls in a row is given by the factorial of the number of girls. Thus, we have:

\[ \text{Total arrangements} = 8! = 40320 \]

### Step 2: Calculate the arrangements where the two sisters are together
To find the arrangements where the two sisters are sitting together, we can treat the two sisters as a single entity or block. This means we now have 7 entities to arrange: the block of sisters and the 6 other girls.

The number of ways to arrange these 7 entities is:

\[ \text{Arrangements of 7 entities} = 7! = 5040 \]

Since the two sisters can switch places within their block, we also need to multiply by the number of ways to arrange the two sisters:

\[ \text{Arrangements of sisters} = 2! = 2 \]

Thus, the total arrangements where the sisters are together is:

\[ \text{Total arrangements with sisters together} = 7! \times 2! = 5040 \times 2 = 10080 \]

### Step 3: Calculate the arrangements where the sisters are not together
To find the number of arrangements where the sisters are not sitting together, we subtract the number of arrangements where they are together from the total arrangements:

\[ \text{Arrangements where sisters are not together} = 8! - (7! \times 2!) \]

Substituting the values we calculated:

\[ \text{Arrangements where sisters are not together} = 40320 - 10080 = 30240 \]

### Final Answer
Thus, the number of ways in which the girls can sit in a row so that the two sisters are not sitting together is:

\[ \boxed{30240} \]