The letters of the word 'MEERUT' are arranged in all possible ways as in a dictionary, then the rank of the word 'MEERUT' is

Options

A. 119
B. 120
C. 121
D. 122

Explanation

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

### Step 1: Arrange the letters in alphabetical order
The letters of the word 'MEERUT' are E, E, M, R, T, U. Arranging them in alphabetical order gives us:
- E, E, M, R, T, U

### Step 2: Calculate the total arrangements before 'MEERUT'
We will count how many words can be formed that come before 'MEERUT' in a dictionary arrangement.

1. **Words starting with 'E':**
- If 'E' is the first letter, the remaining letters are E, M, R, T, U.
- The number of arrangements of these letters is given by:
\[ \frac{5!}{2!} = \frac{120}{2} = 60 \]
(We divide by \(2!\) because there are two E's.)

2. **Words starting with 'M':**
- The next letter is 'M', so we fix 'M' as the first letter. The remaining letters are E, E, R, T, U.
- Now we consider the second letter:
- If the second letter is 'E':
- The remaining letters are E, R, T, U.
- The number of arrangements is:
\[ 4! = 24 \]
- If the second letter is 'R':
- The remaining letters are E, E, T, U.
- The number of arrangements is:
\[ \frac{4!}{2!} = \frac{24}{2} = 12 \]
- If the second letter is 'T':
- The remaining letters are E, E, R, U.
- The number of arrangements is:
\[ \frac{4!}{2!} = \frac{24}{2} = 12 \]
- If the second letter is 'U':
- The remaining letters are E, E, R, T.
- The number of arrangements is:
\[ \frac{4!}{2!} = \frac{24}{2} = 12 \]

### Step 3: Count the arrangements
Now we sum up all the arrangements:
- From the first letter 'E': 60
- From the first letter 'M' and second letter 'E': 24
- From the first letter 'M' and second letter 'R': 12
- From the first letter 'M' and second letter 'T': 12
- From the first letter 'M' and second letter 'U': 12

Total arrangements before 'MEERUT':
\[ 60 + 24 + 12 + 12 + 12 = 120 \]

### Step 4: Add the position of 'MEERUT'
Now we need to consider the word 'MEERUT' itself. Since we have counted all arrangements before it, we add 1 for 'MEERUT':
\[ 120 + 1 = 121 \]

### Final Answer
The rank of the word 'MEERUT' is **121**.

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