5 girls and 10 boys sit at random in a row having 15 chairs numbered as 1 to 15. Find the probability that end seats are occupied by the girls and between only two girls odd number of boys sit.

Options

A. `(20)/(100)`
B. `(20)/(3003)`
C. `(10)/(1001)`
D. none of these

Answer

To solve the problem step by step, we need to find the probability that the end seats are occupied by girls and that there are only two girls with an odd number of boys sitting between them.

### Step 1: Understanding the Arrangement
We have 5 girls and 10 boys, making a total of 15 people. The arrangement must satisfy two conditions:
1. The end seats (1 and 15) must be occupied by girls.
2. Between the two girls at the ends, there must be an odd number of boys.

### Step 2: Fixing the Girls in the End Seats
Let's denote the girls as G1, G2, G3, G4, and G5. We can choose any 2 girls to occupy the end seats (positions 1 and 15). The number of ways to choose 2 girls from 5 is given by the combination formula:

\[ \text{Number of ways to choose 2 girls} = \binom{5}{2} = \frac{5!}{2!(5-2)!} = \frac{5 \times 4}{2 \times 1} = 10 \]

### Step 3: Arranging the Remaining Girls and Boys
After placing 2 girls in the end seats, we have 3 girls left and 10 boys to arrange in the remaining 13 seats (positions 2 to 14).

### Step 4: Positioning Boys Between Girls
Since we need an odd number of boys between the two girls at the ends, the possible odd counts of boys that can fit between them are 1, 3, 5, 7, or 9.

Let’s denote the number of boys between the two girls as \( b \). The total number of boys is 10, so the remaining boys after placing \( b \) boys between the girls is \( 10 - b \).

### Step 5: Counting Valid Arrangements
For each valid odd \( b \):
- If \( b = 1 \): We have 1 boy between the girls and 9 boys left.
- If \( b = 3 \): We have 3 boys between the girls and 7 boys left.
- If \( b = 5 \): We have 5 boys between the girls and 5 boys left.
- If \( b = 7 \): We have 7 boys between the girls and 3 boys left.
- If \( b = 9 \): We have 9 boys between the girls and 1 boy left.

### Step 6: Arranging Boys and Girls
For each arrangement:
1. Choose the number of boys \( b \) to place between the two girls.
2. Arrange the remaining boys and the 3 girls in the remaining seats.

The total arrangements for each case can be calculated as follows:
- The number of ways to arrange the remaining boys and girls in the remaining seats is given by the factorial of the total number of people left.

### Step 7: Total Arrangements
The total number of arrangements for each case can be summed up.

### Step 8: Total Outcomes
The total number of ways to arrange all 15 people (5 girls and 10 boys) without any restriction is:

\[ 15! = 1,307,674,368,000 \]

### Step 9: Calculating the Probability
The probability is given by the ratio of the number of favorable outcomes to the total outcomes:

\[ P = \frac{\text{Number of favorable outcomes}}{15!} \]

### Final Calculation
After calculating the number of favorable arrangements for each case and summing them up, we will divide by \( 15! \) to find the final probability.