A total of 6 Boys and 6 girls are to sit in a row alternatively and in a circle. Let m be the number of arrangements in the row and n be the number of arrangements in the circle. If `k=(m)/(10_(n))`, then the value of k is

Answer

To solve the problem, we need to calculate the number of arrangements of 6 boys and 6 girls sitting alternatively in a row and in a circle. We will denote the number of arrangements in a row as \( m \) and in a circle as \( n \). Finally, we will find the value of \( k = \frac{m}{10n} \).

### Step 1: Calculate \( m \) (Arrangements in a Row)

1. **Arrangement in a Row**: Since the boys and girls must sit alternately, we can start with either a boy or a girl.
- If we start with a boy, the arrangement will be: B G B G B G B G B G B G.
- If we start with a girl, the arrangement will be: G B G B G B G B G B G B.

2. **Calculating Arrangements**:
- There are 6 boys and 6 girls. The number of ways to arrange the boys is \( 6! \) and the number of ways to arrange the girls is also \( 6! \).
- Since we can start with either a boy or a girl, we multiply by 2.

Therefore, the total number of arrangements in a row is:
\[ m = 2 \times 6! \times 6! \]

### Step 2: Calculate \( n \) (Arrangements in a Circle)

1. **Arrangement in a Circle**: In circular arrangements, we fix one person to avoid counting rotations as different arrangements. We can fix one boy in one position.

2. **Calculating Arrangements**:
- After fixing one boy, we have 5 remaining boys and 6 girls to arrange.
- The arrangement will be: B G B G B G B G B G B G.
- The number of ways to arrange the remaining 5 boys in a circle is \( (6-1)! = 5! \).
- The number of ways to arrange the 6 girls is \( 6! \).

Therefore, the total number of arrangements in a circle is:
\[ n = 5! \times 6! \]

### Step 3: Calculate \( k \)

Now, we can substitute \( m \) and \( n \) into the formula for \( k \):
\[ k = \frac{m}{10n} = \frac{2 \times 6! \times 6!}{10 \times 5! \times 6!} \]

### Step 4: Simplify \( k \)

1. Cancel \( 6! \) from the numerator and denominator:
\[ k = \frac{2 \times 6!}{10 \times 5!} \]

2. Recall that \( 6! = 6 \times 5! \):
\[ k = \frac{2 \times 6 \times 5!}{10 \times 5!} \]

3. Cancel \( 5! \):
\[ k = \frac{2 \times 6}{10} = \frac{12}{10} = \frac{6}{5} \]

4. Convert to decimal:
\[ k = 1.2 \]

### Final Answer:
The value of \( k \) is \( \frac{6}{5} \) or \( 1.2 \).

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