The number of words which can be formed out of the letters of the word `ARTICLE`, so that vowels occupy the even place is

Options

A. 1440
B. 144
C. `9!`
D. `10!`

Answer

To solve the problem of finding the number of words that can be formed from the letters of the word "ARTICLE" such that vowels occupy the even places, we can follow these steps:

### Step 1: Identify the letters in the word "ARTICLE"
The word "ARTICLE" consists of 7 letters: A, R, T, I, C, L, E.

### Step 2: Count the vowels and consonants
- **Vowels**: A, I, E (3 vowels)
- **Consonants**: R, T, C, L (4 consonants)

### Step 3: Determine the positions for vowels and consonants
In a 7-letter word, the positions are as follows:
1. 1st position (odd)
2. 2nd position (even)
3. 3rd position (odd)
4. 4th position (even)
5. 5th position (odd)
6. 6th position (even)
7. 7th position (odd)

The even positions are 2, 4, and 6. Thus, we will place the 3 vowels in these 3 even positions.

### Step 4: Arrange the vowels in the even positions
The number of ways to arrange 3 vowels (A, I, E) in the 3 even positions (2, 4, 6) is given by:
\[ 3! = 6 \text{ ways} \]

### Step 5: Arrange the consonants in the odd positions
The odd positions are 1, 3, 5, and 7. We have 4 consonants (R, T, C, L) to place in these positions. The number of ways to arrange 4 consonants in the 4 odd positions is given by:
\[ 4! = 24 \text{ ways} \]

### Step 6: Calculate the total arrangements
The total number of arrangements of the letters such that vowels occupy the even positions is the product of the arrangements of vowels and consonants:
\[ \text{Total arrangements} = 3! \times 4! = 6 \times 24 = 144 \]

### Final Answer
Thus, the total number of words that can be formed from the letters of the word "ARTICLE" such that vowels occupy the even places is **144**.

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