Let m denotes the number of ways in which 5 boys and 5 girls can be arranged in a line alternately and n denotes the number of ways in which 5 boys and 5 girls an be arranged in a circle so that no two boys are together . If m= kn then the value of k is :

Options

A. 2
B. 5
C. 6
D. 10

Answer

To solve the problem, we need to find the values of \( m \) and \( n \) based on the arrangements of boys and girls, and then determine the value of \( k \) such that \( m = kn \).

### Step 1: Calculate \( m \) (Arrangement in a Line Alternately)
We have 5 boys and 5 girls. The arrangement must be such that boys and girls alternate. There are two possible patterns for this arrangement:
1. Boy - Girl - Boy - Girl - Boy - Girl - Boy - Girl - Boy - Girl
2. Girl - Boy - Girl - Boy - Girl - Boy - Girl - Boy - Girl - Boy

For each of these patterns, we can arrange the boys and girls independently.

- The number of ways to arrange 5 boys is \( 5! \).
- The number of ways to arrange 5 girls is \( 5! \).

Since there are 2 patterns, the total number of arrangements \( m \) is given by:
\[ m = 2 \times (5!) \times (5!) \]

### Step 2: Calculate \( n \) (Arrangement in a Circle with No Two Boys Together)
When arranging in a circle, we first fix the position of the girls to avoid rotations being counted multiple times.

1. The number of ways to arrange 5 girls in a circle is \( (5-1)! = 4! \).
2. Once the girls are arranged, there are 5 gaps created between them (including the gap after the last girl) where the boys can be placed.

To ensure no two boys are together, we can place one boy in each gap. The number of ways to arrange 5 boys in these gaps is \( 5! \).

Thus, the total number of arrangements \( n \) is given by:
\[ n = (4!) \times (5!) \]

### Step 3: Set Up the Equation \( m = kn \)
We know:
\[ m = 2 \times (5!) \times (5!) \]
\[ n = (4!) \times (5!) \]

We can substitute these into the equation \( m = kn \):
\[ 2 \times (5!) \times (5!) = k \times (4!) \times (5!) \]

### Step 4: Simplify the Equation
We can divide both sides by \( (5!) \) (assuming \( 5! \neq 0 \)):
\[ 2 \times (5!) = k \times (4!) \]

Now, we can express \( k \):
\[ k = \frac{2 \times (5!)}{(4!)} \]

### Step 5: Calculate \( k \)
We know that \( 5! = 5 \times 4! \), so:
\[ k = \frac{2 \times (5 \times 4!)}{(4!)} = 2 \times 5 = 10 \]

### Final Answer
Thus, the value of \( k \) is:
\[ \boxed{10} \]