5 boys and 5 girls are sitting in a row randomly . The probability that boys and girls sits alternatively , is

Options

A. `(5)/(126)`
B. `(1)/(42)`
C. `(4)/(126)`
D. `(1)/(126)`

Answer

To solve the problem of finding the probability that 5 boys and 5 girls sit alternately in a row, we can follow these steps:

### Step 1: Calculate the total arrangements of 10 people
The total number of ways to arrange 10 people (5 boys and 5 girls) in a row is given by the factorial of the total number of people.

\[ \text{Total arrangements} = 10! \]

### Step 2: Determine the arrangements for alternate seating
To have boys and girls sitting alternately, there are two possible patterns:
1. Boys sit first: B G B G B G B G B G
2. Girls sit first: G B G B G B G B G B

### Step 3: Calculate arrangements for each pattern
For each pattern, there are 5 boys and 5 girls. The boys can be arranged among themselves in \(5!\) ways, and the girls can also be arranged among themselves in \(5!\) ways.

\[ \text{Ways for boys first} = 5! \times 5! \]
\[ \text{Ways for girls first} = 5! \times 5! \]

### Step 4: Combine the arrangements
Since there are two patterns (boys first and girls first), we can add the arrangements from both cases:

\[ \text{Total favorable arrangements} = 5! \times 5! + 5! \times 5! = 2 \times (5! \times 5!) \]

### Step 5: Calculate the probability
The probability that boys and girls sit alternately is given by the ratio of the number of favorable arrangements to the total arrangements.

\[ \text{Probability} = \frac{\text{Total favorable arrangements}}{\text{Total arrangements}} = \frac{2 \times (5! \times 5!)}{10!} \]

### Step 6: Simplify the expression
We know that \(10! = 10 \times 9 \times 8 \times 7 \times 6 \times 5!\). Therefore, we can simplify the probability:

\[ \text{Probability} = \frac{2 \times (5! \times 5!)}{10 \times 9 \times 8 \times 7 \times 6 \times 5!} = \frac{2 \times 5!}{10 \times 9 \times 8 \times 7 \times 6} \]

### Step 7: Calculate \(5!\)
Calculating \(5!\):

\[ 5! = 120 \]

### Step 8: Substitute and calculate the final probability
Substituting \(5!\) into the probability expression:

\[ \text{Probability} = \frac{2 \times 120}{10 \times 9 \times 8 \times 7 \times 6} \]

Calculating the denominator:

\[ 10 \times 9 \times 8 \times 7 \times 6 = 30240 \]

So,

\[ \text{Probability} = \frac{240}{30240} = \frac{1}{126} \]

Thus, the final answer is:

\[ \text{Probability} = \frac{1}{126} \]