In 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 words can be formed from the letters of the word "RAINBOW" such that the first and last letters are vowels, we can follow these steps:

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

### Step 2: Determine the possible combinations for the first and last letters
We need to choose two vowels for the first and last positions. The possible combinations are:
1. First letter A, Last letter I
2. First letter A, Last letter O
3. First letter I, Last letter A
4. First letter I, Last letter O
5. First letter O, Last letter A
6. First letter O, Last letter I

This gives us a total of 6 combinations for the first and last letters.

### Step 3: Count the remaining letters
After selecting the first and last letters, we have 5 letters left from the original 7 letters in "RAINBOW".

### Step 4: Calculate the arrangements of the remaining letters
The remaining letters will be arranged in the middle positions. Since we have 5 letters left, we can arrange these 5 letters in 5! (5 factorial) ways.

Calculating 5!:
\[ 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \]

### Step 5: Calculate the total number of arrangements
Since there are 6 ways to choose the first and last letters and 120 ways to arrange the remaining letters, we multiply these together to get the total number of arrangements.

Total arrangements = Number of ways to choose first and last letters × Arrangements of remaining letters
\[ = 6 \times 120 = 720 \]

### Final Answer
Thus, the total number of words in which the first and last letters are vowels is **720**.

---