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

Options

A. 360
B. 44
C. 72
D. 54

Answer

To find the number of arrangements of the letters in 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 6 letters: E, A, M, C, E, T.
- 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 3 consonants is given by:
\[ 3! = 6 \]

### Step 3: Create gaps for vowels
Once the consonants are arranged, we will create gaps to place the vowels. The arrangement of the consonants (M, C, T) creates 4 gaps:
- Before M
- Between M and C
- Between C and T
- After T

This can be visualized as:
\[ \_ M \_ C \_ T \_ \]
Thus, we have 4 gaps to place the vowels.

### Step 4: Choose gaps 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 in the selected gaps. Since the letter E is repeated, the number of arrangements of the vowels is given by:
\[ \frac{3!}{2!} = 3 \]

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

Thus, the total number of arrangements of the letters in the word "EAMCET" such that no two vowels are adjacent is **72**.

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