Problem 29 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

How many ways are there to arrange the letters in the word GARDEN with vowels in alphabetical order[2004] (a) 480 (b) 240 (c) 360 (d) 120

Step-by-Step Solution

Answer

There are 360 ways to arrange the letters with vowels in alphabetical order.

Step 1: Identify the Vowels and Consonants

The word "GARDEN" consists of 6 letters: G, A, R, D, E, N. Among these, the vowels are A and E, while the consonants are G, R, D, and N.

Step 2: Alphabetize the Vowels

Since the vowels need to be in alphabetical order in the arrangement, A must come before E in each valid arrangement. Thus, the possible order for vowels is fixed as "AE."

Step 3: Total Arrangements of Letters

Calculate the total number of arrangements of the 6 distinct letters in "GARDEN" without any restrictions. This is given by the factorial of the number of letters: \[ 6! = 720. \]

Step 4: Adjust for Vowel Arrangement Restriction

Since the arrangement of vowels "AE" is fixed and needs no permutation, the number of favorable arrangements is reduced by the different ways vowels could be arranged among themselves if not fixed. This involves multiplying by the number of ways to arrange the 2 vowels if they were not fixed: \[ 2! = 2. \]Thus, divide the total possible arrangements by this number,\[ \frac{6!}{2} = \frac{720}{2} = 360. \]

Step 5: Final Calculation for Restricted Arrangement

Verify that the correct calculation accounts for keeping A always before E, which means \[ \frac{720}{2} = 360 \] is indeed the correct number of arrangements with A always coming before E.

Key Concepts

Factorial Calculation

Factorial calculation is a fundamental concept in combinatorics, which involves computing the number of ways to arrange a set of items. When dealing with permutations, we use the factorial of a number to determine the possible arrangements of a list of distinct objects. The factorial of a number, denoted as \( n! \), is the product of all positive integers less than or equal to \( n \). For example, the factorial of 6, written as \( 6! \), is calculated as:

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

This value, 720, represents the total number of ways to arrange 6 different letters or items. Understanding factorial calculation is crucial for solving permutation problems, especially when specific restrictions are imposed on arrangements. By learning how to use factorials, you can easily solve many different types of ordering problems.

Vowel Arrangement

The concept of vowel arrangement comes into play when specific elements in a set need to follow a particular order. In our exercise, the vowels in the word "GARDEN"—which are 'A' and 'E'—must be arranged alphabetically as 'AE'.To handle this vowel arrangement condition, first recognize that fixing the order of vowels essentially reduces the complexity of the problem. This is because once the order 'AE' is set, it does not change regardless of how many times you shuffle the consonants or other letters around it. Next, let's explore why these restrictions affect the total number of arrangements. Normally, if we did not have to worry about any order, the vowels A and E would have 2 possible arrangements ('AE' or 'EA'). To accommodate this requirement and adhere to the alphabetical order, only one arrangement is permissible: 'AE'. So, the adjustment is done by dividing the total arrangements by the factorial of the number of vowels, which calculates as \( 2! = 2 \). By restricting and fixing vowel arrangements, we simplify the solution to many permutation exercises.

Combinatorics

Combinatorics is a branch of mathematics that deals with counting, arrangement, and combination of objects. It plays a significant role in solving permutation problems like the one presented. In our exercise, we need to arrange the letters of 'GARDEN' with vowels in a specific order. Utilizing combinatorics helps in understanding and calculating these arrangements.When faced with such problems, we start by determining the total number of arrangements using factorial calculations, as covered earlier. Then, when limitations or specific patterns are required, as with our vowel arrangement, combinatorics principles guide us on how to adjust or narrow down potential combinations.

• First, calculate all possible permutations using \( 6! \).
• Next, apply restrictions like vowel placement using additional calculations, often involving division by factorial.

Apart from arranging items in order, combinatorics can also help in choosing a subset of items without concern for order, known as "combinations." Understanding these principles and how they apply to permutation problems allows for effectively tackling various exercises, from simple arrangements to complex conditional ones.