How Many Distinct Permutations Can Be Made From the Letters of the Word 'Infinity' When Vowels Are Never Together?

The word infinity consists of 9 letters, comprising 3 vowels (i, i, i) and 6 consonants (n, n, f, t, y). The challenge here is to find how many distinct permutations can be created such that the vowels are never together. This problem requires a systematic approach to consider the positions of both vowels and consonants while ensuring no three vowels are adjacent.

Step-by-Step Analysis

First, let's focus on the consonants, which form the backbone of the word. There are 5 consonants in the word: n, n, f, t, y. Notably, the letter 'n' appears twice.

Step 1: Arranging the Consonants

The number of distinct permutations of these consonants, considering the repetition of 'n', is calculated by the formula for permutations of multiset elements:

[frac{5!}{2!} frac{120}{2} 60]

Step 2: Creating Gaps for Vowels

After arranging the consonants, we have 6 potential gaps where vowels can be placed. These gaps are created as follows:

Before the first consonant, Between the first and second consonants, Between the second and third consonants, Between the third and fourth consonants, Between the fourth and fifth consonants, After the last consonant.

Step 3: Placing the Vowels in Non-Adjacent Gaps

Since no two vowels can be adjacent, we must choose 3 out of these 6 gaps to place the three 'i's. The number of ways to select these 3 gaps from 6 is given by the combination formula:

[_{6}C_{3} frac{6!}{3!(6-3)!} frac{720}{6 times 6} 20]

Step 4: Calculating the Total Number of Distinct Permutations

The total number of distinct permutations is thus the product of the number of ways to arrange the consonants and the number of ways to place the vowels in the non-adjacent gaps:

[60 times 20 1,200]

Conclusion

In summary, there are 1,200 distinct permutations of the letters in the word 'infinity' where the vowels are never together. This methodical approach of first arranging consonants and then selecting appropriate gaps for the vowels ensures a thorough and accurate count.

Related Keywords and Terms

Keywords: permutations, infinity, vowel arrangements