Forming Words with VARUNJOSHI: Maintaining Vowel and Consonant Relative Positions

The problem of forming words using the letters of the word 'VARUNJOSHI,' while maintaining the relative positions of vowels and consonants, involves a careful analysis of combinatorial mathematics. This article explores the number of unique words that can be created under these constraints, using the word 'VARUNJOSHI' as an example.

Identifying Vowels and Consonants

Firstly, let us identify the vowels and consonants in the given word 'VARUNJOSHI'. The vowels are A, U, O, I, and the consonants are V, R, N, J, S, H. The positions of these vowels and consonants can be denoted as follows:

Vowels: 2, 4, 7, 9 (A, U, O, I)Consonants: 1, 3, 5, 6, 8, 10 (V, R, N, J, S, H)

These positions will remain constant in any word formed, ensuring the relative positions of vowels and consonants are maintained.

Arranging Vowels and Consonants

The arrangement of vowels and consonants is independent of each other. Let's analyze the possible arrangements for each set.

Vowel Arrangement

The vowels A, U, O, I can be arranged in the identified positions (2, 4, 7, 9). The number of ways to arrange 4 distinct vowels is given by the factorial of 4, denoted as 4!:

4! 4 × 3 × 2 × 1 24

Consonant Arrangement

The consonants V, R, N, J, S, H can be arranged in the identified positions (1, 3, 5, 6, 8, 10). The number of ways to arrange 6 distinct consonants is given by the factorial of 6, denoted as 6!:

6! 6 × 5 × 4 × 3 × 2 × 1 720

Total Arrangements

Since the arrangements of vowels and consonants are independent, the total number of arrangements of the word 'VARUNJOSHI' is the product of the individual arrangements:

Total arrangements 4! × 6! 24 × 720 17280

Conclusion

Therefore, the total number of unique words that can be formed using the letters of 'VARUNJOSHI' while maintaining the relative positions of vowels and consonants is 17280.

This calculation provides an in-depth understanding of the combinatorial possibilities within a constrained set of letter replacements and arrangements. The method described can be adapted to solve similar problems where the relative positions of certain characters must be maintained.