In how many ways can the letters of the word RAINBOW be arranged so that only two vowels always remain together ?

Options

A. 2880
B. 1440
C. 3200
D. none of these

Answer

To solve the problem of arranging the letters of the word "RAINBOW" such that only two vowels always remain together, we can follow these steps:

### Step 1: Identify the vowels and consonants
The word "RAINBOW" consists of 7 letters: R, A, I, N, B, O, W. Among these, the vowels are A, I, and O.

### Step 2: Group two vowels together
Since we want only two vowels to remain together, we can consider two vowels as a single unit. Let's take A and I as one unit (let's denote it as X). The remaining letters are R, N, B, O, and W.

Now we have the following units to arrange:
- X (which represents A and I together)
- O
- R
- N
- B
- W

This gives us a total of 6 units to arrange: X, O, R, N, B, W.

### Step 3: Calculate arrangements of the units
The number of ways to arrange these 6 units is given by 6! (factorial of 6):
\[ 6! = 720 \]

### Step 4: Arrange the vowels within the unit
Now, within the unit X (which is A and I), we can arrange the two vowels in 2! (factorial of 2) ways:
\[ 2! = 2 \]

### Step 5: Combine the arrangements
Now, we multiply the arrangements of the units by the arrangements of the vowels:
\[ Total arrangements = 6! \times 2! = 720 \times 2 = 1440 \]

### Step 6: Consider the third vowel
Since we have to ensure that only two vowels are together, we need to place the third vowel (O) in the arrangement. The third vowel O can be placed in any of the 7 positions (before the first letter, between any two letters, or after the last letter).

### Step 7: Calculate the total arrangements with O
Since O can be placed in any of the 7 positions, we need to multiply the previous total by 7:
\[ Total arrangements with O = 1440 \times 7 = 10080 \]

### Final Answer
Thus, the total number of ways to arrange the letters of the word "RAINBOW" such that only two vowels always remain together is **10080**.

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