Question 1177980
Let's break down this problem step-by-step.

Understanding the Problem

Total Cars: N
Our Car: Parked among N cars, not at either end.
Remaining Cars: r (including our car)
Goal: Find the probability that both neighboring places are empty.
1. Total Possible Arrangements

We know that r spots are occupied, including our car. Since our car's position is fixed, we need to choose the remaining r-1 spots from the N-1 available spots (excluding our car).

Total ways to choose r-1 spots from N-1 spots: C(N-1, r-1) = (N-1)! / [(r-1)! * (N-r)!]
2. Favorable Arrangements

We want to find the arrangements where our car's two neighboring spots are empty.

Fix Our Car: Our car's position is fixed.

Empty Neighbors: The two spots next to our car must be empty.

Remaining Spots: We have N - 3 remaining spots (excluding our car and its neighbors).

Remaining Cars: We need to choose r - 1 spots from the N - 3 spots.

Favorable ways to choose r-1 spots from N-3 spots: C(N-3, r-1) = (N-3)! / [(r-1)! * (N-r-2)!]

3. Calculate the Probability

The probability is the ratio of favorable arrangements to total arrangements:

Probability = C(N-3, r-1) / C(N-1, r-1)
Probability = [(N-3)! / ((r-1)! * (N-r-2)!)] / [(N-1)! / ((r-1)! * (N-r)!)]
Probability = [(N-3)! / (N-r-2)!] * [(N-r)! / (N-1)!]
Probability = [(N-3)! / (N-1)!] * [(N-r)! / (N-r-2)!]
Probability = [1 / ((N-1)(N-2))] * [(N-r)(N-r-1)]
Probability = [(N-r)(N-r-1)] / [(N-1)(N-2)]
Therefore, the probability that both neighboring places are empty is [(N-r)(N-r-1)] / [(N-1)(N-2)].