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 alternately in both a row and a circle. We will denote the number of arrangements in a row as \( m \) and in a circle as \( n \). We will then find the value of \( k = \frac{m}{10n} \).

### Step 1: Calculate the arrangements in a row (m)

1. **Arrangements in a row**: Since we need to arrange 6 boys and 6 girls alternately, we can start with a boy in the first position. The arrangement will look like this:
\[ B G B G B G B G B G B G \]
This means we have 6 positions for boys and 6 positions for girls.

2. **Permutations**: The number of ways to arrange 6 boys is \( 6! \) and the number of ways to arrange 6 girls is also \( 6! \). Therefore, the total arrangements when starting with a boy is:
\[ 6! \times 6! \]

3. **Starting with a girl**: If we start with a girl, the arrangement will look like this:
\[ G B G B G B G B G B G B \]
The number of arrangements remains the same, \( 6! \times 6! \).

4. **Total arrangements in a row**: Since we can start with either a boy or a girl, we multiply the arrangements by 2:
\[ m = 2 \times (6! \times 6!) = 2 \times 6! \times 6! \]

### Step 2: Calculate the arrangements in a circle (n)

1. **Arrangements in a circle**: When arranging in a circle, we can fix one person to eliminate the effect of rotations. Let's fix one boy in position.

2. **Remaining positions**: After fixing one boy, we have 5 boys and 6 girls left to arrange. The arrangement will look like this:
\[ B G B G B G B G B G \]
Now we have 5 boys and 6 girls to arrange in the remaining positions.

3. **Permutations**: The number of ways to arrange the remaining 5 boys is \( 5! \) and the number of ways to arrange 6 girls is \( 6! \). Therefore, the total arrangements in a circle when starting with a boy is:
\[ 5! \times 6! \]

4. **Starting with a girl**: If we fix a girl instead, we will have the same number of arrangements:
\[ 5! \times 6! \]

5. **Total arrangements in a circle**: Since we can start with either a boy or a girl, we multiply the arrangements by 2:
\[ n = 2 \times (5! \times 6!) = 2 \times 5! \times 6! \]

### Step 3: Calculate k

1. **Substituting values**: Now we substitute \( m \) and \( n \) into the equation for \( k \):
\[ k = \frac{m}{10n} = \frac{2 \times 6! \times 6!}{10 \times (2 \times 5! \times 6!)} \]

2. **Simplifying**: The \( 2 \) cancels out:
\[ k = \frac{6! \times 6!}{10 \times 5! \times 6!} = \frac{6!}{10 \times 5!} \]

3. **Using factorial properties**: Since \( 6! = 6 \times 5! \):
\[ k = \frac{6 \times 5!}{10 \times 5!} = \frac{6}{10} = \frac{3}{5} \]

4. **Final value**: Thus, the value of \( k \) is:
\[ k = 1.2 \]

### Final Answer:
The value of \( k \) is \( 1.2 \).