Problem 48 - A Complete Resource Book in Mathematics for JEE Main

Question

There are \(n\) persons \((n \geq 3)\), among whom are \(A\) and \(B\), who are made to stand in a row in random order. Probability that there is exactly one person between \(A\) and \(B\) is (A) \(\frac{n-2}{n(n-1)}\) (B) \(\frac{2(n-2)}{n(n-1)}\) (C) \(2 / n\) (D) none os these

Step-by-Step Solution

Answer

Option (B): \(\frac{2(n-2)}{n(n-1)}\).

Step 1: Determine Total Arrangements

First, consider the total number of ways to arrange the \(n\) people in a line. Since each person can take any of the \(n\) positions, the total number of arrangements is \(n!\).

Step 2: Position Selection for A and B

To ensure there is exactly one person between \(A\) and \(B\), consider \(A\) taking position 1, meaning \(B\) must be in position 3, or if \(A\) is in position 2, then \(B\) could be in position 4, and so forth. This pattern continues until \(A\) is in position \(n-2\) with \(B\) in position \(n\).

Step 3: Count Possible Arrangements for A and B

There are \(n - 2\) positions \(A\) can take that allow \(B\) to have exactly one person between them. Moreover, since \(B\) could be before \(A\) or after \(A\), we double this to account for both directional possibilities, giving us \(2(n-2)\) ways to have exactly one person between them.

Step 4: Arrangement of Remaining Persons

After choosing positions for \(A\) and \(B\), there are \(n - 2\) persons left to arrange in the remaining \(n\) positions. The number of arrangements of these \(n - 2\) persons in \(n - 2\) positions is \((n-2)!\).

Step 5: Calculate Favorable Outcomes

Combine the counted arrangements of \(A\) and \(B\) with the arrangements of the remaining people: \(2(n-2) \times (n-2)!\).

Step 6: Calculate the Required Probability

The probability of having exactly one person between \(A\) and \(B\) is given by the ratio of favorable outcomes to total outcomes: \[\frac{2(n-2)(n-2)!}{n!}\]. Simplify this expression: \[ \frac{2(n-2)}{n(n-1)} \].

Step 7: Identify the Correct Option

Comparing with the provided options, the correct probability is option \(B\): \(\frac{2(n-2)}{n(n-1)}\).

Key Concepts

Permutations

Permutations are all about arranging a set of objects in a specific order. In probability problems that deal with permutations, like the one in our exercise, the focus is on how objects like people or letters can be ordered.

• Let's say you have a list of people, and you want to see all possible ways they can line up.
• If you have 3 people, they can be arranged in 3! ("3 factorial") ways, which equals 6.

This is because the first person has 3 options, the second 2 options left, and the third person has only 1 option remaining. For our problem, permutations help us find the total number of arrangements of the group of people, which is given by the formula for factorial, denoted as \(n!\).
Understanding permutations is key, as it helps determine the underlying possibilities within a problem.

Combinatorics

Combinatorics involves counting and arranging objects, which is essential in problems like the one given. It includes both permutations and combinations, but here we're focused on scenarios that involve arranging items.

• It helps in figuring out the specific conditions, such as having exactly one person between two others, which introduces restrictions on how we count the arrangements.
• Using combinatorics, we can figure out how to position person A and person B, with exactly one person between them, and how many such arrangements exist.

This idea extends from simply counting all arrangements to selecting specific cases that meet the requirements of being "favorable." Thanks to combinatorics, we understand why there are \((n-2)\) positions they could be in while meeting the specific condition. This concept tells us how we count these specific favorable outcomes cleverly and correctly.

Mathematical Reasoning

Mathematical reasoning is the glue that binds all steps together in solving such probability problems. It's about logically following through each step to ensure each calculation is meaningful and correct.

• First, establish the total number of possible arrangements.
• Next, logically work out how many ways the specific condition of "one person between A and B" can be satisfied.

This involves breaking down the steps:
1. Determine how many scenarios meet the condition by accounting for A and B in potential "correct" positions.
2. Arrange the remaining people after accounting for A and B.

Good mathematical reasoning ensures the thoughtful combination of favorable outcomes and a comparison with all outcomes to find the correct probability, like option B in this problem. This reasoning allows students to make informed decisions and arrive at solutions systematically, which is crucial in both simple and complex probability problems.