Arranging the Letters of 'PATLIPUTRA' While Maintaining Consonant and Vowel Positions

Introduction

Understanding the arrangement of the letters in a word while maintaining the relative positions of vowels and consonants can be a fascinating mathematical problem. The word 'PATLIPUTRA' offers a unique challenge as it contains several repeated letters. This article will explore various techniques to determine the number of ways to arrange the letters of 'PATLIPUTRA' while keeping the relative positions of vowels and consonants intact.

Step-by-Step Analysis of 'PATLIPUTRA'

Step 1: Identify Consonants and Vowels

The word 'PATLIPUTRA' consists of 10 letters, with specific occurrences of consonants and vowels.

Consonants: p (2 times) t (2 times) l r Vowels: a (2 times) i u

Step 2: Fixed Positions

The problem requires that the relative positions of consonants and vowels must remain unchanged. According to the given arrangement, the consonants occupy the 1st, 3rd, 6th, and 8th positions, while the vowels occupy the 2nd, 5th, 7th, and 10th positions.

Step 3: Arranging Consonants

For the consonants, we have a total of 6 letters, including 2 'p's and 2 't's. To calculate the number of distinct arrangements, we use the formula for permutations of multiset:

Number of arrangements frac{6!}{2!cdot2!}

First, calculate 6! (6 factorial) and 2! (2 factorial)

6! 720

2! 2

Therefore, the number of arrangements of consonants is:

Number of arrangements of consonants frac{720}{2cdot2} 180

Step 4: Arranging Vowels

Now, let's consider the vowels. The 4 vowels are 'a' (2 times), 'i', and 'u'. Again, we use the formula for permutations of multiset:

Number of arrangements frac{4!}{2!}

Calculate 4! and 2! first:

4! 24

2! 2

Therefore, the number of arrangements of vowels is:

Number of arrangements of vowels frac{24}{2} 12

Step 5: Total Arrangements

To find the total number of arrangements, multiply the number of arrangements of consonants by the number of arrangements of vowels:

Total arrangements 180 times; 12 2160

This means there are 2160 distinct ways to arrange the letters of 'PATLIPUTRA' while maintaining the relative positions of consonants and vowels.

Alternative Method

Another way to solve this problem is to consider the selection of positions for vowels and consonants. Since 4 positions out of 10 are designated for vowels and 6 for consonants, we can use combinations:

Number of ways to select 4 positions for vowels out of 10 positions {10 choose 4} 210

This method confirms the arrangement calculation by choosing the positions for vowels first, then filling those positions with the given instances of vowels and consonants.

Conclusively, the total number of arrangements of the letters in 'PATLIPUTRA' while keeping the relative positions of consonants and vowels is 2160. Both methods yield the same result, providing a thorough understanding of the problem.