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

Options

A. 7!
B. 30
C. 180
D. `4! xx 5!`

Answer

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

### Step 1: Identify the letters and their types
The word "PARTICLE" consists of 8 letters: P, A, R, T, I, C, L, E. Among these, the vowels are A, I, and E, and the consonants are P, R, T, C, and L.

**Hint:** Remember to categorize the letters into vowels and consonants.

### Step 2: Determine the positions for vowels
In the word "PARTICLE," the even positions are 2, 4, 6, and 8. This gives us 4 even positions to fill with vowels.

**Hint:** List the positions clearly to avoid confusion later.

### Step 3: Choose positions for the vowels
We need to choose 3 out of these 4 even positions to place the vowels. The number of ways to choose 3 positions from 4 is given by the combination formula \( \binom{n}{r} \), where \( n \) is the total number of positions and \( r \) is the number of positions to choose.

\[ \text{Number of ways to choose 3 positions from 4} = \binom{4}{3} = 4 \]

**Hint:** Use the combination formula \( \binom{n}{r} = \frac{n!}{r!(n-r)!} \).

### Step 4: Arrange the vowels in the chosen positions
Once we have chosen 3 positions for the vowels, we can arrange the 3 vowels (A, I, E) in these positions. The number of arrangements of 3 vowels is given by \( 3! \).

\[ \text{Number of arrangements of vowels} = 3! = 6 \]

**Hint:** Remember that the arrangement of distinct items is calculated using factorial.

### Step 5: Arrange the consonants in the remaining positions
After placing the vowels, we have 5 remaining positions (1, 3, 5, 7) for the consonants (P, R, T, C, L). The number of ways to arrange 5 consonants in these 5 positions is given by \( 5! \).

\[ \text{Number of arrangements of consonants} = 5! = 120 \]

**Hint:** Factorial is used for arranging distinct items in available positions.

### Step 6: Calculate the total number of arrangements
To find the total number of arrangements where vowels occupy the even positions, we multiply the number of ways to choose the positions for the vowels, the arrangements of the vowels, and the arrangements of the consonants.

\[ \text{Total arrangements} = \binom{4}{3} \times 3! \times 5! = 4 \times 6 \times 120 \]

Calculating this gives:

\[ 4 \times 6 = 24 \]
\[ 24 \times 120 = 2880 \]

### Final Answer
The total number of words that can be formed from the letters of the word "PARTICLE" such that the vowels occupy the even places is **2880**.

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