Calculating 5-Letter Combinations with Specific Consonants and Vowels

In this article, we will explore how to calculate the number of unique 5-letter arrangements from the letters in STAT MEET given that each arrangement must contain exactly 3 consonants and 2 vowels. This problem involves a combination of combinatorial mathematics and permutation principles, making it a perfect case study for understanding complex word arrangements.

The Problem Statement

We are given the string "STAT MEET". It contains the following unique letters and their occurrences:

Consonants:

S: 1 T: 2 (T, T) M: 1 (M)

Vowels:

A: 1 (A) E: 3 (E, E, E)

The task is to determine the number of different 5-letter combinations where exactly 3 consonants and 2 vowels are used.

Step 1: Identify Consonants and Vowels

From the given letters, we have:

Consonants:

S (1) T (2) M (1)

So, the unique consonants are S, T, and M.

Vowels:

A (1) E (3)

So, the unique vowels are A and E.

Step 2: Choose 3 Consonants and 2 Vowels

We need to choose 3 consonants from S, T, and M, and 2 vowels from A and E.

Choosing Consonants:

S, T, M S, T, T T, T, M

Therefore, there are 3 ways to choose the 3 consonants.

Choosing Vowels:

A, E E, E

So, there are 2 ways to choose the 2 vowels.

The total number of ways to choose 3 consonants and 2 vowels is:

Total combinations 3 (consonants) times; 2 (vowels) 6 ways

Step 3: Combine Choices and Calculate Arrangements

Now, we need to calculate the number of arrangements for each combination of 3 consonants and 2 vowels. Let's go through each combination:

S, T, T, A, E: 5! / 2! 120 / 2 60 S, T, M, A, E: 5! 120 T, T, M, A, E: 5! / 2! 120 / 2 60 S, T, T, E, E: 5! / (2! times; 2!) 120 / 4 30 S, T, M, E, E: 5! / 2! 120 / 2 60 T, T, M, E, E: 5! / (2! times; 2!) 120 / 4 30

Step 4: Summing Up the Arrangements

The total number of arrangements is the sum of the arrangements for each combination:

Total arrangements: 60 120 60 30 60 30 360

Final Answer

The total number of different 5-letter arrangements with exactly 3 consonants and 2 vowels from the letters in "STAT MEET" is:

360

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Conclusion: Understanding the principles of combinations and permutations can help solve complex word arrangement problems. This article provides a detailed example of how to calculate the number of 5-letter arrangements with specific requirements, enhancing your problem-solving skills and SEO strategies.