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 vowels and consonants
The word "PENCIL" consists of the letters: P, E, N, C, I, L.
- Vowels: E, I
- Consonants: P, N, C, L

### Step 2: Calculate the total arrangements of the letters
The total number of arrangements of the letters in "PENCIL" can be calculated using the factorial of the number of letters:
\[ \text{Total arrangements} = 6! = 720 \]

### Step 3: Calculate arrangements where vowels are together
To find the arrangements where the vowels E and I are together, we can treat the pair (EI) as a single unit. Thus, we will have the following units to arrange:
- Units: (EI), P, N, C, L
This gives us a total of 5 units to arrange.

The number of arrangements of these 5 units is:
\[ 5! = 120 \]

Since the vowels can be arranged among themselves (EI can be either EI or IE), we multiply by the arrangements of the vowels:
\[ \text{Arrangements with vowels together} = 5! \times 2! = 120 \times 2 = 240 \]

### Step 4: Calculate arrangements where no two vowels are together
Now, we can find the arrangements where no two vowels are together by subtracting the arrangements where the vowels are together from the total arrangements:
\[ \text{Arrangements with no two vowels together} = \text{Total arrangements} - \text{Arrangements with vowels together} \]
\[ = 720 - 240 = 480 \]

### Final Answer
The number of different words that can be formed from the letters of the word "PENCIL" such that no two vowels are together is **480**.

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