Arranging the Word INTERMEDIATE with Vowels in Alphabetical Order

When faced with arranging a specific word like INTERMEDIATE with certain constraints, a logical and systematic approach is crucial. The problem at hand is to determine in how many ways the word INTERMEDIATE can be rearranged so that all the vowels are in alphabetical order. Let’s explore the solution through a step-by-step method that leverages combinatorial mathematics to ensure accuracy.

Understanding the Constraints

The word INTERMEDIATE consists of twelve letters containing six vowels (I, E, E, A, I, A) and six consonants (N, T, R, M, T, D). The challenge is to arrange the letters such that the vowels appear in the order A, E, E, I, I, A. This is a classic problem that can be divided into two parts: first, arranging the consonants, and then placing the vowels in the remaining spots while maintaining their sequence.

Arranging the Consonants

Let's begin by analyzing the consonants in the word. There are six consonants: N, T, R, M, T, D. To start, we can consider arranging these consonants in any order and then fill in the gaps with vowels. The first step is to choose 6 positions out of 12 for the consonants. This can be done using the combination formula 12C6, which calculates the number of ways to choose 6 positions from 12 without considering the order.

12C6 can be calculated as follows:

[ 12C6 frac{12!}{6! times 6!} ]

However, note that the consonants include two 'T's and one 'N', 'R', 'M', and 'D'. To account for the repetition of the 'T', the number of distinct permutations of the consonants is given by:

[ frac{6!}{2!} ]

Therefore, the total number of ways to arrange the consonants is:

[ 12C6 times frac{6!}{2!} ]

Placing the Vowels

Once the consonants are arranged, we have six remaining spots for the vowels. Given that the vowels need to be in alphabetical order (A, E, E, I, I, A), the only valid arrangement in the available spots is:

A in one of the spots Three Es in the next three spots Two Is in the last two spots

Since the order of the vowels is fixed, there is only one way to place them once the consonants are arranged.

Combining the Steps

Combining the two parts, the total number of ways to arrange the word INTERMEDIATE such that all vowels are in alphabetical order is:

[ 12C6 times frac{6!}{2!} frac{12!}{6!6!} times frac{6!}{2!} frac{12!}{6!2!} ]

This provides us with the exact count of arrangements with all vowels in alphabetical order.

Conclusion

By carefully considering the constraints and applying combinatorial principles, we can accurately determine the number of ways to arrange the letters of the word INTERMEDIATE with vowels in alphabetical order. The process involves first arranging the consonants and then placing the vowels in reserved positions, ensuring that the vowels maintain their alphabetical order.

Keywords: Permutations, Vowels in Alphabetical Order, Word Arrangements