Problem 102 - A Complete Resource Book in Mathematics for JEE Main 2017

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

(B) \(\frac{1}{7}\).

Step 1: Understanding the circular permutation

In a round table arrangement with 15 people, we fix one person to break the symmetry of the circle. This leaves us with 14 people to arrange around this circle.

Step 2: Positioning A and B with 4 people between them

To have exactly 4 people sitting between A and B, treat A and B as markers. If we fix A's position, there are two possible scenarios (clockwise or counterclockwise) as B can either be to the left or right with 4 people between them. There are only 2 such positions around one fixed A.

Step 3: Counting the number of favorable arrangements

From Step 2, there are 2 favorable ways to place B relative to A.

Step 4: Calculating total arrangements of the remaining people

With A fixed, 13 other people are free to be placed in the remaining 13 positions. This can be done in \(13!\) ways.

Step 5: Calculating the probability

The total ways to arrange 14 people (considering one is fixed) is \(14!\). The probability is therefore computed as the fraction of favorable outcomes over total outcomes: \[ \text{Probability} = \frac{2 \times 13!}{14!} = \frac{2}{14} = \frac{1}{7}. \]

Step 6: Conclusion

Thus, the probability that there are 4 persons between A and B in the round table arrangement is \(\frac{1}{7}\).

Key Concepts

Probability Calculation

Probability is a branch of mathematics that deals with the likelihood or chance of different outcomes. In many real-world scenarios, probability is used to predict how likely an event is to occur. When calculating probability, we use the following formula:

• The probability of an event = \( \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}} \).

For instance, in the original problem where we must determine the probability of specific seating arrangements, it's crucial to first identify the favorable outcomes. In this case, it would be the specific ways in which people can be seated around a circular table.

After pinpointing the favorable outcomes, dividing this by the total number of possible arrangements gives us the probability. Such calculations always require a detailed breakdown of the problem, as seen with deciding how many people are between certain individuals at a table. This method ensures accuracy and clarity.

Factorial in Combinatorics

Factorial is an essential concept in combinatorics, which is the branch of mathematics focused on counting, arrangement, and combination. The factorial of a number \( n \), denoted as \( n! \), is the product of all positive integers less than or equal to \( n \).

• For example, \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \).

Understanding the factorial concept simplifies the calculation of permutations and combinations, as it provides a quick way to calculate the total arrangements or selections from a set.

In the context of circular permutations like in our round table problem, factorial is used to calculate both the total number of ways people can be seated (14!) and the number of ways left when one person's position is fixed (13!). This helps in analyzing the potential arrangements or layouts in a problem efficiently.

Knowing when to use factorials can greatly enhance your problem-solving skills in probability and combinatorics, making these calculations more intuitive.

Round Table Problem

The round table problem is a classic permutation problem that involves arranging people or objects in a circle. In circular permutation, the focus is on positions relative to each other, unlike linear permutations where edges matter. To understand this concept, imagine you have people sitting in a circle.

To calculate permutations in a round table problem, we often fix one person's position to break the rotational symmetry. This effectively linearizes the circle, allowing calculations akin to linear permutations. This is because there are no distinct ends in a circle; hence fixing a position avoids overcounting equivalent arrangements.

• For example, if you have 15 people, you fix one person, and calculate how to arrange the 14 remaining people, leading to \( 14! \) arrangements.

The round table problem often involves placing specific conditions, like having a certain number of people between two specified individuals. By fixing one person's position and counting the valid seating configurations for the rest, we find solutions to such problems efficiently.