If all the letters of the word QUEUE are arranged in all manner as they are in a dictionary, then the rank of the word QUEUE is: a) 15th b) 16th c) 17th d) 18th

To determine the rank of the word "QUEUE" when all its letters are arranged in dictionary order, we need to systematically calculate the total number of permutations that precede it.

### Step 1: Understand the letters in "QUEUE"
The word "QUEUE" consists of the letters: Q, U, E, U, E.
There are 5 letters, with E appearing twice and U appearing twice.

### Step 2: Arrange letters in dictionary order
In alphabetical order, the letters are: E, E, Q, U, U.

### Step 3: Permutations starting with each letter
Let us compute the number of words starting with each letter until we reach "QUEUE".

#### Case 1: Words starting with "E"
If the first letter is "E", the remaining letters are: E, Q, U, U.
The total number of permutations of these letters is:
\[ \text{Permutations} = \frac{4!}{2! \cdot 2!} = 6 \]
This means 6 words start with "E" and precede "QUEUE".

#### Case 2: Words starting with "Q"
If the first letter is "Q", the remaining letters are: E, E, U, U.
The total number of permutations of these letters is:
\[ \text{Permutations} = \frac{4!}{2! \cdot 2!} = 6 \]
This means 6 words start with "Q".

#### Case 3: Words starting with "QE"
If the first two letters are "QE", the remaining letters are: E, U, U.
The total number of permutations of these letters is:
\[ \text{Permutations} = \frac{3!}{2!} = 3 \]
This means 3 words start with "QE".

#### Case 4: Words starting with "QUE"
If the first three letters are "QUE", the remaining letters are: E, U.
The total number of permutations of these letters is:
\[ \text{Permutations} = \frac{2!}{1!} = 2 \]
This means 2 words start with "QUE".

#### Case 5: Words starting with "QUEUE"
The word "QUEUE" itself is the next word.

### Step 4: Calculate the rank of "QUEUE"
- Words starting with "E": 6
- Words starting with "Q" but before "QUE": 6
- Words starting with "QE": 3
- Words starting with "QUE" but before "QUEUE": 2
- "QUEUE" is the next word.

The rank of "QUEUE" is:
\[ 6 + 6 + 3 + 2 + 1 = 18 \]

### Final Answer
The rank of the word "QUEUE" is:
d) 18th