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

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 places, we can follow these steps:

### Step 1: Identify the letters in the word "PARTICLE"
The word "PARTICLE" consists of 8 letters: P, A, R, T, I, C, L, E.

### Step 2: Identify the vowels and consonants
In "PARTICLE," the vowels are A, I, and E. Therefore:
- Number of vowels = 3 (A, I, E)
- Number of consonants = 5 (P, R, T, C, L)

### Step 3: Determine the even positions
In an 8-letter word, the even positions are 2, 4, 6, and 8. Thus, there are 4 even positions available.

### Step 4: Choose positions for the vowels
We need to place the 3 vowels in the 4 even positions. The number of ways to choose 3 positions from 4 is given by the combination formula:
\[ \text{Number of ways to choose positions} = \binom{4}{3} = 4 \]

### Step 5: Arrange the vowels in the chosen positions
The 3 vowels can be arranged among themselves in the chosen positions. The number of arrangements of 3 vowels is given by:
\[ \text{Number of arrangements of vowels} = 3! = 6 \]

### Step 6: Calculate the total arrangements for vowels
The total arrangements for the vowels in the even positions is:
\[ \text{Total arrangements for vowels} = \binom{4}{3} \times 3! = 4 \times 6 = 24 \]

### Step 7: Arrange the consonants
Now, we have 5 consonants (P, R, T, C, L) that need to occupy the remaining 5 positions (1, 3, 5, 7). The number of arrangements of the 5 consonants is given by:
\[ \text{Number of arrangements of consonants} = 5! = 120 \]

### Step 8: Calculate the total arrangements
Finally, the total number of arrangements of the letters in the word "PARTICLE" such that the vowels occupy the even places is:
\[ \text{Total arrangements} = \text{Arrangements of vowels} \times \text{Arrangements of consonants} = 24 \times 120 = 2880 \]

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

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