Problem 63 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

If eight persons are to address a meeting then the number of ways in which a specified speaker is to speak before another specified speaker, is (A) 40320 (B) 2520 (C) 20160 (D) None of these

Step-by-Step Solution

Answer

The number of ways is 20160.

Step 1: Understanding the Problem

We need to determine the number of ways to arrange eight speakers in such a way that a specified speaker (Speaker A) speaks before another specified speaker (Speaker B). This is a classic problem of permutations with a condition.

Step 2: Calculate Total Permutations

First, we calculate the total number of ways to arrange all eight speakers without any restrictions. This is given by the factorial of 8, denoted as \(8!\). So, \(8! = 40320\).

Step 3: Set Condition on Speakers

We need only those permutations where Speaker A speaks before Speaker B. When arranging two specific items (Speaker A and Speaker B), there are two possible orders: A before B or B before A. Given this specific condition, only half of the total arrangements will have A before B.

Step 4: Calculate Valid Arrangements

Since there are two possibilities (A before B or B before A) and we want A before B, divide the total permutations by 2. Thus, the number of permutations where A speaks before B will be \( \frac{8!}{2} = \frac{40320}{2} = 20160\).

Key Concepts

Factorial

In mathematics, the factorial of a number is a fundamental concept often denoted by an exclamation mark, such as in "8!". This operation signifies the product of all positive integers up to that number. For example, the factorial of 8 (written as 8!) is calculated by multiplying all the integers from 1 to 8 together:

• 8! = 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1
• This results in 40320

The factorial function grows very quickly, making it incredibly useful in problems involving permutations and combinations. It helps in counting the total number of ways to arrange a set of objects, which is crucial for solving problems in combinatorics.

Conditional Arrangements

Conditional arrangements refer to permutations that are organized under specific conditions or restrictions. In this exercise, one such condition is that one specified speaker must speak before another. This means we are concerned only with those arrangements of eight speakers that satisfy this specific order.

• First, determine the total number of arrangements without any conditions — this is calculated through the factorial of the total number of items (8! in this instance).
• Next, apply the condition: half of these arrangements will naturally have Speaker A before Speaker B, and the other half will have the reverse order (B before A).

By interpreting the condition on our arrangements, we see that they are essentially divided equally among the two potential sequences of interest. Thus, we take the total number and divide it by 2, leading to 20160 ways that meet the condition.

Combinatorics

Combinatorics is a fascinating field of mathematics focusing on counting, arrangement, and combinations of objects. Problems in this domain often involve determining the number of ways to arrange or select items, which is exactly the task at hand in this exercise.

• Permutations are a central theme in combinatorics. They deal with the arrangement of objects where the order is important.
• Factorial calculations are a basis for determining permutations, providing a way to quantify total possible arrangements.
• Conditional arrangements narrow these permutations by applying specific sequence rules, as seen when determining how many speaker setups place one before another.

The principles of combinatorics enable us to solve complex problems by simplifying them into smaller, more manageable components, such as using the factorial function and applying conditions to filter the desired outcomes.