In how many ways can the letters of the world CORPORATION be arranged so that vowels always occupy even places?

Options

A. 120
B. 2700
C. 720
D. 7200

Answer

To solve the problem of arranging the letters of the word "CORPORATION" such that the vowels always occupy even places, we can follow these steps:

### Step 1: Identify the letters and their types
The word "CORPORATION" consists of 11 letters:
- Vowels: O, O, O, A, I (5 vowels)
- Consonants: C, R, P, R, T, N (6 consonants)

### Step 2: Determine the positions for vowels
In the arrangement of 11 letters, the even positions are 2, 4, 6, 8, and 10. This gives us a total of 5 even positions available for the vowels.

### Step 3: Arrange the vowels in the even positions
We need to arrange the 5 vowels (O, O, O, A, I) in the 5 even positions. Since the letter 'O' is repeated 3 times, the number of arrangements of the vowels is calculated using the formula for permutations of multiset:

\[ \text{Arrangements of vowels} = \frac{5!}{3!} = \frac{120}{6} = 20 \]

### Step 4: Arrange the consonants in the remaining positions
The remaining positions (1, 3, 5, 7, 9, 11) can be filled with the 6 consonants (C, R, P, R, T, N). The letter 'R' is repeated 2 times, so the number of arrangements of the consonants is:

\[ \text{Arrangements of consonants} = \frac{6!}{2!} = \frac{720}{2} = 360 \]

### Step 5: Calculate the total arrangements
To find the total arrangements of the letters in the word "CORPORATION" with the given condition, we multiply the number of arrangements of vowels by the number of arrangements of consonants:

\[ \text{Total arrangements} = (\text{Arrangements of vowels}) \times (\text{Arrangements of consonants}) = 20 \times 360 = 7200 \]

### Final Answer
The total number of ways to arrange the letters of the word "CORPORATION" such that the vowels always occupy even places is **7200**.

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