Two girls and four boys any made to sit in a line in such a way that the girls should not sit together. By how many ways this is possible?

Options

A. 720
B. 480
C. 360
D. 240

Answer

To solve the problem of arranging two girls and four boys in a line such that the girls do not sit together, we can follow these steps:

### Step 1: Calculate the total arrangements of boys and girls without any restrictions.
First, we need to find the total number of ways to arrange the 6 individuals (2 girls and 4 boys). The total number of arrangements is given by the factorial of the total number of individuals:

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

### Step 2: Calculate the arrangements where the girls sit together.
To find the arrangements where the girls sit together, we can treat the two girls as a single unit or block. This means we now have 5 units to arrange: the block of girls (GG) and the four boys (B1, B2, B3, B4).

The number of ways to arrange these 5 units is:

\[ \text{Arrangements with girls together} = 5! = 120 \]

However, within the block of girls, the two girls can also be arranged among themselves. The number of arrangements of the two girls is:

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

Thus, the total arrangements where the girls sit together is:

\[ \text{Total arrangements with girls together} = 5! \times 2! = 120 \times 2 = 240 \]

### Step 3: Calculate the arrangements where the girls do not sit together.
To find the arrangements where the girls do not sit together, we subtract the arrangements where the girls sit together from the total arrangements:

\[ \text{Arrangements with girls not together} = \text{Total arrangements} - \text{Arrangements with girls together} \]

Substituting the values we calculated:

\[ \text{Arrangements with girls not together} = 720 - 240 = 480 \]

### Final Answer:
Thus, the number of ways to arrange two girls and four boys in a line such that the girls do not sit together is **480**.

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