If the letters of the word VERMA are arranged in all possible ways and these words are written out as in a dictionary, then the rank of the word VERMA is :

Options

A. 108
B. 117
C. 810
D. 180

Answer

To find the rank of the word "VERMA" when all permutations of its letters are arranged in alphabetical order, we can follow these steps:

### Step-by-Step Solution:

1. **List the Letters in Alphabetical Order**:
The letters in "VERMA" are A, E, M, R, V. Arranging them in alphabetical order gives us: A, E, M, R, V.

2. **Count Words Starting with Each Letter Before V**:
- **Words starting with A**: The remaining letters are E, M, R, V. The number of permutations is \(4! = 24\).
- **Words starting with E**: The remaining letters are A, M, R, V. The number of permutations is \(4! = 24\).
- **Words starting with M**: The remaining letters are A, E, R, V. The number of permutations is \(4! = 24\).
- **Words starting with R**: The remaining letters are A, E, M, V. The number of permutations is \(4! = 24\).

Total words starting with A, E, M, and R:
\[ 24 + 24 + 24 + 24 = 96 \]

3. **Count Words Starting with V**:
Now, we consider words that start with V. The remaining letters are A, E, M, R.

- **Words starting with VA**: The remaining letters are E, M, R. The number of permutations is \(3! = 6\).
- **Words starting with VE**: The remaining letters are A, M, R. The number of permutations is \(3! = 6\).
- **Words starting with VM**: The remaining letters are A, E, R. The number of permutations is \(3! = 6\).
- **Words starting with VR**: The remaining letters are A, E, M. The number of permutations is \(3! = 6\).

Total words starting with VA, VE, VM, and VR:
\[ 6 + 6 + 6 + 6 = 24 \]

4. **Count Words Starting with VE**:
Now we look at words starting with VE. The remaining letters are A, M, R.

- **Words starting with VEA**: The remaining letters are M, R. The number of permutations is \(2! = 2\).
- **Words starting with VEM**: The remaining letters are A, R. The number of permutations is \(2! = 2\).
- **Words starting with VER**: The remaining letters are A, M. The number of permutations is \(2! = 2\).

Total words starting with VEA, VEM, and VER:
\[ 2 + 2 + 2 = 6 \]

5. **Count the Rank of VERMA**:
Finally, we reach the word "VERMA". The words that come before "VERMA" in alphabetical order are:
- Words starting with A: 24
- Words starting with E: 24
- Words starting with M: 24
- Words starting with R: 24
- Words starting with VA: 6
- Words starting with VE: 6 (before "VERMA")
- Words starting with VEA: 2
- Words starting with VEM: 2

Adding these gives us:
\[ 24 + 24 + 24 + 24 + 6 + 6 + 2 + 2 = 112 \]

Since "VERMA" is the next word, its rank is:
\[ 112 + 1 = 113 \]

### Final Answer:
The rank of the word "VERMA" is **113**.