If all the letters of the word 'QUEST' are arranged in all possible ways and put in dictionary order, then find the rank of the given word

Answer

To find the rank of the word "QUEST" when all the letters are arranged in dictionary order, we can follow these steps:

### Step 1: List the letters in alphabetical order
The letters in "QUEST" are Q, U, E, S, T. Arranging these in alphabetical order gives us:
- E, Q, S, T, U

### Step 2: Count the arrangements starting with letters before 'Q'
We need to find how many words can be formed starting with letters that come before 'Q' in alphabetical order.

1. **Words starting with 'E':**
- Remaining letters: Q, U, S, T
- The number of arrangements = 4! = 24

2. **Words starting with 'Q':**
- Now we consider words starting with 'Q'. We will check the next letters.

### Step 3: Count the arrangements starting with 'QU'
Next, we check words starting with 'QU'.

1. **Words starting with 'QU':**
- Remaining letters: E, S, T
- The number of arrangements = 3! = 6

### Step 4: Count the arrangements starting with 'QUES'
Next, we check words starting with 'QUES'.

1. **Words starting with 'QUES':**
- Remaining letters: T
- The number of arrangements = 1! = 1 (only "QUEST")

### Step 5: Calculate the total rank
Now, we can sum up all the arrangements:
- Words starting with 'E': 24
- Words starting with 'Q' and 'E': 6
- Words starting with 'Q', 'U', 'E': 1 (which is "QUEST")

Total = 24 (from 'E') + 6 (from 'QU') + 1 (from 'QUES') = 31

### Step 6: Determine the rank of "QUEST"
The rank of "QUEST" is the total number of arrangements before it plus one (for "QUEST" itself):
- Rank = 31 + 1 = 32

### Final Answer
The rank of the word "QUEST" is **32**.

---