Problem 102 - A Complete Resource Book in Mathematics for JEE Main

Question

Fifteen persons, among whom are \(A\) and \(B\), sit down at random at a round table. The probability that there are 4 persons between \(A\) and \(B\) is (A) \(\frac{1}{3}\) (B) \(\frac{1}{7}\) (C) \(\frac{1}{5}\) (D) none of these

Step-by-Step Solution

Answer

The probability is \(\frac{1}{7}\), meaning option (B) is correct.

Step 1: Understand the Arrangement

In a circular arrangement of 15 people, one person can be fixed and the remaining can be arranged around them. So, we fix one person at a spot and the others can be arranged in a circle. Therefore, there are 14 factorial ways to arrange the remaining 14 persons.

Step 2: Determine Possible Positions for B

We are looking for the number of ways person B can be positioned such that there are exactly 4 people between A and B. Since persons A and B are distinct, and the arrangement is circular, B can be placed either 5 seats clockwise or 5 seats counterclockwise from A.

Step 3: Calculate Number of Favorable Cases

For person B to be placed exactly 5 seats away from A in a circular arrangement, we have 2 possibilities: either to the left or to the right (clockwise or counterclockwise). This gives us 2 possible favorable arrangements for B.

Step 4: Calculate Total Arrangements

After fixing person A, there are 13 remaining people that can be arranged in any order around the table, giving us \(13!\) (factorial of 13) possible total arrangements.

Step 5: Compute the Probability

The probability is the ratio of favorable cases to total arrangements. That is, \(\frac{2}{13!}\) favorable arrangements divided by the total possible \(13!\) arrangements, which simplifies to \(\frac{2}{13}\).

Key Concepts

Circular Permutation

When it comes to arranging objects or people in a circular manner, we refer to this as circular permutation. Unlike linear permutations where the order clearly starts and ends, circular permutations require considering one item or person as a reference point. This is because, in a circle, rotating the arrangement does not change the order. For instance, if we have 15 people sitting around a round table, we fix one person as a reference point and then arrange the remaining 14 people around them.
This `fixing` simplifies our problem. It allows us to calculate the arrangements as \( (n-1)! \) which represents the remaining people. Basically, in our solution, by fixing one person (e.g., person A), we end up arranging the others in \( 14! \) ways. This approach is pivotal in solving circular permutation problems such as the one in the exercise.

Combinatorics

Combinatorics is a branch of mathematics that deals with counting, arrangement, and combination of objects. In the context of our problem, it plays a crucial role in calculating the possible arrangements.
Firstly, we determine how person B can be arranged relative to person A. Since A and B need exactly 4 people between them, B can be placed 5 seats away in either direction - clockwise or counterclockwise. This is a classic example of combinatorial calculation where we consider different positioning options. Through combinatorics, we count two favorable arrangements of B concerning A. These contribute to finding the probability of person B being five seats apart from A in a circle, which exemplifies the practical application of combinatorics.

Factorial Calculation

Factorial calculation is a fundamental concept in permutations, especially when dealing with circular permutations. A factorial, denoted by \( n! \), is the product of all positive integers up to \( n \). It tells us the number of ways to arrange \( n \) items linearly.
In our problem, after deciding the relative position of B, we need to account for the remaining 13 people around our fixed reference point. The number of such arrangements is estimated by \( 13! \). It’s fascinating because factorial calculations turn the seemingly complex problem of permutations and combinations into a series of multiplications, making it manageable.Understanding factorials helps illuminate why \( 13! \) represents the total arrangements and why the favorable probability (\( \frac{2}{13} \)) is derived from dividing possible favorable arrangements by \( 13! \). Factorials thus play an indispensable role in not just counting arrangements but also in solving problems systematically.