Arranging the Word 'MAGIC' with Vowels Together

In this article, we delve into the process of arranging the letters of the word MAGIC such that the vowels A and I always appear together. This problem involves understanding permutations and combinations, specifically focusing on treating a subset of characters as a single unit. We'll analyze the solution step-by-step using mathematical reasoning and a final check with the J programming language.

Step-by-Step Solution

Step 1: Treating Vowels as a Single Unit

The word 'MAGIC' comprises two vowels, A and I, and three consonants M, G, and C. To ensure the vowels are always together, we can form a single unit of the vowels. This reduces our problem to arranging the units: AI, M, G, and C.

Step 2: Calculating Arrangements of the Units

Since we have 4 units to arrange (AI, M, G, C), the number of possible arrangements is given by the factorial of 4:

$$ 4! 4 times 3 times 2 times 1 24 $$

This step is crucial in determining the total number of unique arrangements when treating the vowels as a single unit.

Step 3: Arranging the Vowels Within Their Block

The block AI (or its reverse IA) can be arranged in 2 different ways. Therefore, we need to multiply the number of arrangements from the previous step by 2:

$$ 2! 2 times 1 2 $$

Step 4: Calculating the Total Arrangements

To find the total number of possible arrangements, we multiply the number of arrangements of the units by the number of arrangements of the vowels within their block:

$$ 4! times 2! 24 times 2 48 $$

Thus, the total number of different arrangements of the word 'MAGIC' where the vowels A and I are always together is 48.

Verification Using J Programming Language

To verify our solution, we can use a J programming language to brute-force the problem. The J language is highly efficient for such tasks and provides a direct way to calculate the number of permutations.

Brute Force Approach

The following J code snippet effectively calculates the number of permutations where vowels A and I are treated as a single unit:

/m.1 1 ssmb 48

This code snippet confirms that the number of different arrangements is 48.

Conclusion

By combining mathematical reasoning and the use of a powerful programming language like J, we have determined that there are 48 distinct arrangements of the word 'MAGIC' where the vowels A and I are always together. This approach can be applied to similar problems involving constraints on character arrangements in words or phrases.

Understanding permutations and combinations is a fundamental skill in various fields, including mathematics, computer science, and data analysis. The ability to solve such problems efficiently is crucial for tasks ranging from optimizing search algorithms to designing secure encryption methods.

Keywords: word arrangement, permutations, vowels together