Permutations of the Word COMMITTEE: An In-Depth Analysis

Permutations play a crucial role in combinatorics and are essential in understanding the intricate nature of certain words. This article delves into the permutations of the word COMMITTEE, focusing on the arrangement of its letters, particularly those starting with a vowel and ending with another vowel. We will explore different patterns and calculate the total number of permutations under these specific conditions.

Overview of the Word COMMITTEE

The word COMMITTEE consists of nine distinct characters: C, O, M, M, I, T, T, E, E. Specifically, it has:

Four vowels (O, I, E, E) Five consonants (C, M, M, T, T)

Understanding Vowel Permutations in COMMITTEE

Let's first understand how the letters in the word COURAGE can be arranged to start with a vowel. The word COURAGE contains 7 distinct letters, with 4 vowels (A, E, O, U). The total number of arrangements for these letters is 7!, which is 5040.

Sub-case 1: Arrangements starting with a Vowel

For each of these 4 vowels, the remaining 6 letters can be arranged in 6! (720) ways. Therefore, the total number of arrangements starting with a vowel is:

$$4 times 6! 2880$$

Alternatively, we can calculate it as the fraction of the total arrangements that start with a vowel. Since 4 out of 7 letters are vowels, the proportion of such arrangements is 4/7. Thus:

$$ frac{4}{7} times 7! frac{4}{7} times 5040 2880 $$

Permutations of COMMITTEE

The word COMMITTEE is a nine-letter word, specifically with the following characteristics:

Four vowels: O, I, E, E Five consonants: C, M, M, T, T

We are interested in the permutations of these letters where the first and last positions are vowels, and there is at least one repetition among the vowels.

Pattern 1: Same Vowel in Both Positions

In this pattern, the first and last positions are occupied by the same vowel, specifically the letter 'E' which is repeated. The number of choices for the vowels in the first and last positions is:

$ binom{2}{2} 1 $

The remaining 7 letters (M, M, T, T, I, O) can be arranged in:

$$ frac{7!}{2! times 2!} frac{5040}{4} 1260 $$

Therefore, the total number of permutations for this pattern is:

$1 times 1260 1260$$

Pattern 2: Two Distinct Vowels in Both Positions

For this pattern, we use the vowels 'I' and 'O', which are distinct. The number of choices for the vowels in the first and last positions is:

$ binom{2}{2} times 2! 2 times 2 4 $

The remaining 7 letters (M, M, T, T, I, O) can be arranged in:

$$ frac{7!}{2! times 2!} 1260 $$

Therefore, the total number of permutations for this pattern is:

$4 times 1260 5040$$

Pattern 3: One Vowel Repeated, and One Vowel Distinct

In this pattern, we have one repeated vowel (either 'E' or 'M') and one distinct vowel (either 'I' or 'O'). The number of choices for the vowels in the first and last positions is:

$ binom{1}{1} times binom{2}{1} times 2! 1 times 2 times 2 4 $

The remaining 7 letters (M, M, T, T, I, O) can be arranged in:

$$ frac{7!}{2! times 2!} 1260 $$

Therefore, the total number of permutations for this pattern is:

$4 times 1260 5040$$

Conclusion

Combining the above patterns, the total number of permutations of the word COMMITTEE where the first and last positions are vowels is:

$1260 5040 5040 11340$$

This analysis shows the versatility of permutations and combinatorial methods in understanding the complexity of word arrangements, especially in cases involving repetitions and specific conditions.