Three boys and three girls are to be seated around a table, in a circle. Among them, the boy X does not want any girl neighbour and the girls Y does not want any boy neighbour. The number of such arrangements possible is

Options

A. A. 4
B. B. 6
C. C. 8
D. D. none of these

Answer

To solve the problem of seating three boys (including boy X) and three girls (including girl Y) around a circular table with the given restrictions, we can follow these steps:

### Step 1: Fixing Boy X's Position
Since we are arranging people in a circle, we can fix one person's position to eliminate the effect of rotations. Let's fix Boy X's position at one point on the table.

**Hint:** Fixing one position helps simplify circular arrangements by converting it into a linear arrangement.

### Step 2: Arranging the Remaining Boys
After fixing Boy X, we have two remaining boys (let's call them Boy 1 and Boy 2). These two boys can be seated in the two positions next to Boy X. The arrangement of these two boys can be done in \(2!\) (factorial of 2) ways.

**Hint:** Factorial notation \(n!\) represents the number of ways to arrange \(n\) distinct items.

### Step 3: Arranging the Girls
Next, we need to place the three girls around the table. Since Girl Y does not want to sit next to Boy X, we can only place the girls in the remaining positions. After placing Boy X and the two boys, there will be three positions left for the girls.

The arrangement of the three girls can be done in \(3!\) (factorial of 3) ways.

**Hint:** Remember that the total arrangements of \(n\) distinct items is given by \(n!\).

### Step 4: Calculating Total Arrangements
Now, we can calculate the total arrangements by multiplying the arrangements of boys and girls:

\[ \text{Total Arrangements} = \text{Arrangements of Boys} \times \text{Arrangements of Girls} = 2! \times 3! \]

Calculating this gives us:

\[ 2! = 2 \quad \text{and} \quad 3! = 6 \]

So,

\[ \text{Total Arrangements} = 2 \times 6 = 12 \]

### Final Answer
Thus, the total number of arrangements possible is **12**.

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