The number of different words that can be formed from the letters of the word 'PENCIL', so that no two vowels are together, is

Options

A. 120
B. 260
C. 240
D. 480

Answer

To solve the problem of finding the number of different words that can be formed from the letters of the word "PENCIL" such that no two vowels are together, we can follow these steps:

### Step 1: Identify the letters and their types
The word "PENCIL" consists of the following letters:
- Consonants: P, C, L (3 consonants)
- Vowels: E, I (2 vowels)

### Step 2: Arrange the consonants
First, we will arrange the consonants. The consonants P, C, and L can be arranged in different ways. The number of arrangements of 3 consonants is given by:

\[ 3! = 6 \]

### Step 3: Determine the available spaces for vowels
Once we arrange the consonants, we need to identify the spaces available for placing the vowels. When the consonants are arranged, they create gaps where vowels can be placed. For example, if we arrange the consonants as "PCL", the arrangement looks like this:

_ P _ C _ L _

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

### Step 4: Choose spaces for the vowels
We need to choose 2 out of these 4 available spaces to place the vowels. The number of ways to choose 2 spaces from 4 is given by the combination formula:

\[ \binom{4}{2} = \frac{4!}{2!(4-2)!} = \frac{4 \times 3}{2 \times 1} = 6 \]

### Step 5: Arrange the vowels
After choosing the spaces, we can arrange the 2 vowels (E and I) in the chosen spaces. The number of arrangements of 2 vowels is:

\[ 2! = 2 \]

### Step 6: Calculate the total arrangements
Now, we can calculate the total number of arrangements by multiplying the number of arrangements of consonants, the number of ways to choose spaces for vowels, and the arrangements of vowels:

\[ \text{Total arrangements} = (3!) \times \left(\binom{4}{2}\right) \times (2!) \]

Substituting the values we calculated:

\[ \text{Total arrangements} = 6 \times 6 \times 2 = 72 \]

### Conclusion
The total number of different words that can be formed from the letters of the word "PENCIL" such that no two vowels are together is **72**.

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