Permutations of the Word 'INDEPENDENCE': An Analysis of Vowel Arrangements

The word 'INDEPENDENCE' presents a unique challenge in combinatorial mathematics, particularly when dealing with permutations and the arrangement of its vowels. This article delves into how many possible arrangements exist for the word 'INDEPENDENCE', with a specific focus on the number of arrangements where the vowels never occur together and instances where all vowels do appear together.

Total Number of Arrangements

The word 'INDEPENDENCE' comprises 12 letters, with the following frequencies: 5 vowels (I, E, E, E, E) and 7 consonants (N, N, N, D, D, C, P). Without any restrictions, the total number of possible arrangements is calculated by:

Formula:

12! / (4! 3! 2!)

12! 479001600

4! 24, 3! 6, 2! 2

Thus, number of all possible arrangements 479001600 / (24 * 6 * 2) 1663200.

Arrangements Where Vowels Never Occur Together

To find the number of arrangements where the vowels do not occur together, we first calculate the total arrangements of the word without any restrictions, which we have already determined to be 1663200. From this, we subtract the number of arrangements where all vowels are together.

Arrangements Where All Vowels Are Together

When all vowels are grouped together, we have a total of 8 characters to arrange: (IEEEE), N, N, N, D, D, C, P. Viewing the vowels as a single unit, we calculate the number of ways to arrange these 8 characters as follows:

Formula:

8! / (3! 2!)

8! 40320

3! 6, 2! 2

Arrangements of 8 characters 40320 / (6 * 2) 3360

Within this single vowel unit, the vowels IEEEE can be arranged in 5! / 4! ways:

Formula:

5! / 4! 5

Therefore, total arrangements where all vowels are together 3360 * 5 16800.

Calculation of Arrangements Where Vowels Never Occur Together

Subtracting the number of arrangements where all vowels are together from the total number of arrangements gives us the number of arrangements where the vowels never occur together:

Total arrangements - Arrangements where all vowels are together 1663200 - 16800 1646400.

Conclusion

To summarize, the word 'INDEPENDENCE' has 1,663,200 possible arrangements if all letters are considered different. When specific conditions such as having the vowels never together or always together are imposed, the number of arrangements can be significantly reduced. For instance, the number of arrangements where all vowels are together is 16,800, and the number of arrangements where vowels never occur together is 1,646,400.