Answer these questions independently of each other
In the different arrangements of the word RAINBOW, how many words are there in which first and last letters are vowels?

Options

A. 360
B. 720
C. 1440
D. 2880

Answer

To solve the problem of how many arrangements of the word "RAINBOW" have vowels as the first and last letters, we can follow these steps:

### Step 1: Identify the vowels in the word "RAINBOW"
The vowels in the word "RAINBOW" are A, I, and O. Thus, we have 3 vowels.

### Step 2: Choose the first and last letters
Since we need the first and last letters to be vowels, we can select the first letter from the 3 vowels and the last letter from the remaining 2 vowels.

- **Choices for the first letter**: 3 (A, I, O)
- **Choices for the last letter**: 2 (after choosing the first letter, only 2 vowels remain)

### Step 3: Calculate the number of ways to choose the first and last letters
The number of ways to choose the first and last letters is:
\[ 3 \text{ (first letter)} \times 2 \text{ (last letter)} = 6 \]

### Step 4: Arrange the remaining letters
After placing the vowels in the first and last positions, we have 5 letters left to arrange. The remaining letters from "RAINBOW" are R, N, B, and W, plus the vowel that was not used.

### Step 5: Calculate the arrangements of the remaining letters
The number of arrangements of these 5 letters is given by \(5!\) (5 factorial), which is:
\[ 5! = 120 \]

### Step 6: Combine the results
To find the total number of arrangements where the first and last letters are vowels, we multiply the number of ways to choose the vowels by the number of arrangements of the remaining letters:
\[ \text{Total arrangements} = 6 \times 120 = 720 \]

### Final Answer
The total number of arrangements of the word "RAINBOW" where the first and last letters are vowels is **720**.

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