Arranging the Letters of 'Monday' Without Vowel Together

The word 'Monday' consists of six distinct letters: M, O, N, D, A, and Y. When arranging these letters, we often encounter scenarios where certain conditions must be met, such as ensuring that the vowels do not appear together. In this article, we will explore how many ways the letters of the word 'Monday' can be arranged such that the vowels are not together.

The Problem and Solution Approach

Firstly, let's identify and count the total number of ways to arrange the letters of the word 'Monday' without any restrictions. Since all six letters are distinct, the total number of arrangements can be calculated as:

6! 720 ways

Approach 1: Vowels Together

An important step in solving this problem is to determine the number of arrangements where the vowels are together. By treating the vowels O and A as a single unit, we use the block OA as a single entity along with the consonants: M, N, D, Y. This gives us 5 units to arrange:

M, N, D, Y, and the block OA or AO

The total number of arrangements of these 5 units is:

5! 120 ways

However, the block OA can be arranged in 2 different ways (either as OA or AO). Therefore, the total number of arrangements where the vowels are together is:

5! x 2 120 x 2 240 ways

Approach 2: Vowels Not Together

To find the number of ways to arrange the letters such that the vowels are not together, we subtract the number of arrangements where the vowels are together from the total number of arrangements:

Arrangements where vowels are not together Total arrangements - Arrangements where vowels are together

720 - 240 480 ways

Conclusion and Summary

Thus, the number of ways to arrange the letters of the word 'Monday' such that the vowels are not together is 480.

Let's summarize the key points:

The total number of arrangements of 'Monday' is 720. The number of arrangements where the vowels are together is 240. The number of arrangements where the vowels are not together is 480.

This method of treating the vowels as a single block and then subtracting the number of arrangements with the block from the total arrangements is a useful strategy in permutation problems.

Mathematically, this solution satisfies the condition of ensuring that the vowels do not appear together while considering all possible permutations of the letters in the word 'Monday'.

Keywords: arrangement, permutation, vowel arrangement