Arranging the Letters of 'Patliputra' while Preserving the Order of Vowels and Consonants

How can the letters of the word 'Patliputra' be arranged such that the relative order of vowels and consonants remains unchanged? This article explores this specific problem and provides a detailed step-by-step approach to solving it using fundamental principles of permutation and combinatorics.

Understanding the Problem

The word 'Patliputra' consists of 10 letters: P, A, T, L, I, P, U, T, R, A. It includes four vowels (A, I, U, A) and six consonants (P, T, L, P, T, R).

The Approach: Selecting Positions for Vowels and Consonants

The problem requires arranging the letters of 'Patliputra' while maintaining the relative positions of vowels and consonants. This means we need to determine the positions of the 4 vowels among the 10 positions. The remaining 6 positions will automatically be assigned to the consonants.

The number of ways to choose 4 positions out of 10 for the vowels (and thus 6 positions for the consonants) is calculated using combinations: [ C(10, 4) frac{10!}{4!(10-4)!} frac{10!}{4!6!} 210 ]

Refining the Approach: Specific Letter Distribution

However, the initial approach didn't consider the specific distribution of letters. The initial allocation of places is simplified, while in fact, we need to take into account the exact count of each character.

Consonants

Given the letter 'P' appears twice, 'T' also appears twice, and 'L' and 'R' each appear once, the number of ways to arrange these consonants is adjusted using the formula for permutations of a multiset:

[ frac{6!}{2!2!} frac{720}{4} 180 ]

Vowels

For the vowels, we have two 'A's, one 'I', and one 'U'. The number of ways to arrange these vowels is:

[ frac{4!}{2!} frac{24}{2} 12 ]

Combining the Results

The total number of ways to arrange the letters of 'Patliputra' while preserving the order of vowels and consonants is the product of the unique arrangements of consonants and vowels:

[ 180 times 12 2160 ]

Conclusion and Further Exploration

This problem highlights the importance of considering multiplicities and specific distributions when dealing with permutations of a multiset. Understanding these concepts is crucial for more complex problems in combinatorial mathematics and for optimizing search algorithms in computer science.

Further reading and exercises can be found in combinatorics texts or online resources, focusing on permutations, combinations, and multiset permutations.

References

Knuth, D. E. (1998). The Art of Computer Programming, Volume 4A: Combinatorial Algorithms, Part 1. Richard A. Brualdi (2010). Introductory Combinatorics Larry J. Goldstein, David I. Schneider, Martha J. Siegel (2013). Finite Mathematics and Its Applications