Problem 41 - A Complete Resource Book in Mathematics for JEE Main

Question

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 theneighbouring places are empty is (A) \(\frac{91}{276}\) (B) \(\frac{15}{184}\) (C) \(\frac{15}{92}\) (D) none of these

Step-by-Step Solution

Answer

The probability is \( \frac{15}{92} \) (Option C).

Step 1: Understanding the Problem

The problem states that a car is parked among 25 cars and the parked car is not at either end. Upon return, 15 places remain occupied. We need to find the probability that both neighboring places of the parked car are empty.

Step 2: Determine Total Outcomes

After parking the car, there are 24 other parking spots. Out of these, 14 spots are occupied since the remaining 15 spots are occupied, including the parked car itself. The total number of ways to occupy 14 out of 24 spots is computed using combinations: \( \binom{24}{14} \).

Step 3: Identify Favorable Outcomes

For the parked car's neighboring positions to be empty, those two specific spots must be among the 10 unoccupied ones. Choose 12 more spots to occupy out of the remaining 22 to ensure the neighbors are empty. This is calculated as \( \binom{22}{12} \).

Step 4: Calculating the Probability

The probability is given by the ratio of favorable outcomes to the total outcomes. Thus, the probability is \( \frac{\binom{22}{12}}{\binom{24}{14}} \). Compute each combination to find the probability.

Step 5: Compute Combinations

Calculate \( \binom{22}{12} \) and \( \binom{24}{14} \). You get \( \binom{22}{12} = 497420 \) and \( \binom{24}{14} = 2496144 \).

Step 6: Calculate and Simplify the Probability

Substitute the results in the probability equation: \( \frac{497420}{2496144} \). Simplify this fraction to find the most reduced terms.

Key Concepts

Combination and Permutation

In probability and statistics, understanding combinations and permutations is crucial. These are mathematical tools that help us determine the number of different ways an event can occur. In this exercise, we focus on combinations because the order of cars being parked doesn't matter.
When dealing with combinations, we use the formula to calculate how many ways we can choose a subset from a larger set:\[\binom{n}{r} = \frac{n!}{r!(n-r)!} \]Here, \( n \) is the total number of items to choose from, and \( r \) is the number of items to choose. The "!" means factorial, which is the product of all positive integers up to that number.

• For our problem, \( n = 24 \) and \( r = 14 \), representing 14 occupied spots out of 24 possible spots.
• Thus, the total number of ways to choose these locations is \( \binom{24}{14} \).

Permutation isn't used in the exercise because the sequence does not affect the outcome.
Understanding the distinction between combination and permutation helps in solving many probability problems.

Occupancy Problem

The occupancy problem deals with distributing a certain number of items into a set of slots. It has a unique twist in our exercise since the challenge is to find the number of arrangements where certain conditions are satisfied.
In this context, the problem is to ensure that the neighboring spots of the parked car are empty, given that some spots remain occupied.
We start with 24 available spots (excluding our parked car's spot), and 14 of these must be filled with other cars.

• First, note that only 10 spots remain unoccupied, so it's crucial that these contain the parked car's neighboring spaces.
• We choose the remaining 12 spots to be occupied from the 22 total spots to ensure that the two empty spots adjacent to the parked car are among the unoccupied.

This concept highlights the importance of constraining a random distribution, focusing on practical conditions like adjacent availability. Understanding how to set conditions in the occupancy problem helps in various similar probability and statistics challenges.

Probability Calculation

Calculation of probabilities is all about determining how likely an event is to happen.This involves comparing favorable outcomes to the total possible outcomes. In this scenario, our task is to compute the probability that both neighboring places to the parked car are empty.
To do this, we use the formula:
\[\text{Probability} = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}\]

• The total number of ways to occupy spots is given by \( \binom{24}{14} \), which includes all possible ways to distribute 14 other cars among 24 spots.
• Meanwhile, the favorable outcomes—in which the spaces adjacent to the parked car are free—are represented by the combination \( \binom{22}{12} \).

Thus, the probability becomes:\[\frac{\binom{22}{12}}{\binom{24}{14}} = \frac{497420}{2496144}\]After simplifying this fraction, we get \[\frac{15}{92}\] which is the chance that both places next to the parked car are empty.
Mastering probability calculations is key in making predictions based on random events.