In the below word how many words are there in which R and W are at the end positions? RAINBOW

Options

A. a. 120
B. b. 180
C. c. 210
D. d. 240

Answer

To solve the problem of how many words can be formed from the letters of the word "RAINBOW" with 'R' and 'W' at the end positions, we can follow these steps:

### Step 1: Identify the total letters in the word
The word "RAINBOW" consists of 7 letters: R, A, I, N, B, O, W.

### Step 2: Fix the positions of R and W
We need to find the arrangements where 'R' and 'W' are at the end positions. There are two cases to consider:
1. R is at the first position and W is at the last position.
2. W is at the first position and R is at the last position.

### Step 3: Calculate arrangements for each case
In both cases, we will have 5 letters left to arrange in the middle positions.

#### Case 1: R at the start and W at the end
- The letters left to arrange are A, I, N, B, O.
- The number of ways to arrange these 5 letters is given by 5! (5 factorial).

#### Case 2: W at the start and R at the end
- The letters left to arrange are A, I, N, B, O.
- The number of ways to arrange these 5 letters is also given by 5! (5 factorial).

### Step 4: Calculate the total arrangements
Since both cases are mutually exclusive, we can add the number of arrangements from both cases:
- Total arrangements = Case 1 + Case 2 = 5! + 5!

### Step 5: Compute the factorial
- 5! = 5 × 4 × 3 × 2 × 1 = 120

### Step 6: Final calculation
- Total arrangements = 120 + 120 = 240

### Conclusion
Thus, the total number of words that can be formed with 'R' and 'W' at the end positions is **240**.

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