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

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

There are 20160 ways for a specified speaker to speak before another specified speaker.

Step 1: Determine the Total Number of Arrangements

First, calculate the total number of possible arrangements for all 8 speakers. The number of different ways to arrange 8 people is given by 8 factorial, which is denoted as \(8!\). Calculate \(8!\) as follows:\[8! = 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 40320\]So, there are 40320 different ways to arrange all 8 speakers.

Step 2: Consider the Order of Two Specific Speakers

Since the problem requires a specific speaker (let's call him Speaker A) to speak before another specific speaker (Speaker B), let's determine how this constrains our arrangements. For any two people, there are two possible orders (A before B, or B before A).

Step 3: Calculate the Required Arrangement

Given that either Speaker A is before Speaker B or Speaker B before Speaker A constitutes half the arrangements, divide the total arrangements by 2:\[\text{Ways with A before B} = \frac{8!}{2} = \frac{40320}{2} = 20160\]Hence, there are 20160 ways in which Speaker A can be ordered before Speaker B among the 8 speakers.

Key Concepts

Factorials

The concept of factorials is fundamental in solving permutation and combination problems. A factorial of a number, denoted as \(n!\), is the product of all positive integers less than or equal to \(n\). Simply put, it's a way to calculate how many different sequences can be arranged within a set number of items. For example:

• \(4! = 4 \times 3 \times 2 \times 1 = 24\)
• \(6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720\)

In this exercise, we calculate the factorial of 8, i.e., \(8!\), which results in 40,320. This serves as the basis to understand the total number of ways all 8 speakers can be arranged. Factorials grow very quickly, reflecting how quickly the number of arrangements increases as more items are involved.

Order Constraints

Order constraints refer to specific conditions or restrictions that influence the order of items. In the context of this exercise, we have a condition where one specified speaker, Speaker A, must speak before another, Speaker B. This constraint changes the number of necessary calculations. Typically, every unique pair of two items can be positioned in two different ways. Consider two speakers: A can either speak before B, or B can speak before A. This creates two choices for every pair.
However, by imposing the condition that Speaker A must go before Speaker B, we limit the choices to just one acceptable scenario out of the two possible for their sequence. So, for every random arrangement of 8 speakers, exactly half will satisfy this specific constraint.

Arrangements

Arrangements in permutations involve determining the sequences in which a set of items can be ordered. This exercise looks at 8 people being arranged in a speaking order. To understand how to find a specific arrangement where one condition must be met, we apply the order constraint.Once the total arrangements (\(8!\) or 40,320) are known, we can apply this constraint. Since we learned from order constraints, half of these arrangements will comply where one speaker, specifically A, precedes another, specifically B.

• Again, divide the total arrangements by 2: \(\frac{8!}{2} = 20,160\).

This means there are 20,160 different ways out of the total 40,320 arrangements where Speaker A can precede Speaker B. Understanding arrangements with constraints helps utilize permutations effectively in problem-solving.