A car is parked by an owner amongst 25 cars in a row, not at either end. On his return he finds that exactly 15 places are still occupied. The probability that both the neighboring places are empty is

Options

A. `(91)/(276)`
B. `(15)/(184)`
C. `(15)/(92)`
D. none of these

Explanation

To solve the problem step by step, we need to find the probability that both neighboring places of the owner's car are empty.

### Step 1: Understand the total number of cars and occupied places
- There are 25 cars in total.
- The owner’s car is parked among these cars, and it is stated that he is not parked at either end.
- Upon returning, he finds that exactly 15 places are still occupied. This includes his own car.

### Step 2: Calculate the number of occupied and empty places
- Since there are 15 occupied places and one of them is the owner’s car, there are 14 other cars parked.
- Therefore, the total number of empty places is:
\[ \text{Total places} - \text{Occupied places} = 25 - 15 = 10 \]

### Step 3: Determine the total number of places excluding the owner’s car
- The owner’s car occupies one place, and since it is not at either end, there are 24 places remaining (excluding the owner's car).

### Step 4: Calculate the total outcomes
- We need to choose 14 cars from these 24 places. The total number of ways to choose 14 cars from 24 is given by the combination formula:
\[ \text{Total outcomes} = \binom{24}{14} \]

### Step 5: Calculate the favorable outcomes
- To find the favorable outcomes where both neighboring places of the owner's car are empty, we need to exclude the two neighboring places.
- If we exclude the two neighboring places, we have:
\[ 24 - 2 = 22 \text{ places} \]
- We still need to park 14 cars in these 22 places. The number of ways to choose 14 cars from these 22 places is:
\[ \text{Favorable outcomes} = \binom{22}{14} \]

### Step 6: Calculate the probability
- The probability that both neighboring places are empty is given by the ratio of favorable outcomes to total outcomes:
\[ P(\text{both neighbors empty}) = \frac{\text{Favorable outcomes}}{\text{Total outcomes}} = \frac{\binom{22}{14}}{\binom{24}{14}} \]

### Step 7: Simplify the probability
- Using the combination formula:
\[ \binom{n}{r} = \frac{n!}{r!(n-r)!} \]
- We can express the probability as:
\[ P = \frac{\frac{22!}{14! \cdot 8!}}{\frac{24!}{14! \cdot 10!}} = \frac{22! \cdot 10!}{24! \cdot 8!} \]
- Simplifying this gives:
\[ P = \frac{22! \cdot 10!}{24 \cdot 23 \cdot 22! \cdot 8!} = \frac{10!}{24 \cdot 23 \cdot 8!} = \frac{10 \cdot 9}{24 \cdot 23} = \frac{90}{552} = \frac{15}{92} \]

### Final Answer
Thus, the probability that both neighboring places are empty is:
\[ \frac{15}{92} \]