All the letters of the word EAMCET are arranged in all possible ways. The number such arrangements in which no two vowels are adjacent to each other is

Options

A. i) 54
B. ii) 72
C. iii) 114
D. iv) 360

Answer

To solve the problem of arranging the letters of the word "EAMCET" such that no two vowels are adjacent, we can follow these steps:

### Step 1: Identify the vowels and consonants
The word "EAMCET" consists of the letters:
- Vowels: E, A, E (3 vowels)
- Consonants: M, C, T (3 consonants)

### Step 2: Arrange the consonants
First, we will arrange the consonants (M, C, T). The number of ways to arrange these 3 consonants is given by:
\[ 3! = 6 \]

### Step 3: Identify the spaces for vowels
When the consonants are arranged, they create gaps where the vowels can be placed. For example, if we arrange the consonants as M, C, T, the arrangement looks like this:
- _ M _ C _ T _

This arrangement has 4 gaps (indicated by underscores) where we can place the vowels.

### Step 4: Choose spaces for the vowels
We need to choose 3 out of these 4 gaps to place the vowels. The number of ways to choose 3 gaps from 4 is given by:
\[ \binom{4}{3} = 4 \]

### Step 5: Arrange the vowels
Next, we need to arrange the vowels E, A, E. Since there are 3 vowels where 'E' is repeated, the number of arrangements of the vowels is given by:
\[ \frac{3!}{2!} = 3 \]
(Here, \(3!\) is the total arrangements of the vowels, and \(2!\) accounts for the repetition of 'E').

### Step 6: Calculate the total arrangements
Now, we can calculate the total arrangements where no two vowels are adjacent:
\[ \text{Total arrangements} = (\text{Arrangements of consonants}) \times (\text{Ways to choose gaps}) \times (\text{Arrangements of vowels}) \]
Substituting the values we calculated:
\[ \text{Total arrangements} = 6 \times 4 \times 3 = 72 \]

### Final Answer
The number of arrangements of the letters in the word "EAMCET" such that no two vowels are adjacent is **72**.

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