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 alphabetical order, we can follow these steps:

### Step 1: Arrange the letters in alphabetical order
The letters of the word "QUEST" are: Q, U, E, S, T.
Arranging them 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 the alphabetical order.

- The first letter in alphabetical order is 'E'. If we fix 'E' as the first letter, the remaining letters are Q, S, T, U. The number of arrangements of these 4 letters is given by \(4!\):
\[ 4! = 24 \]

### Step 3: Fix 'Q' and count arrangements starting with letters before 'U'
Now, we fix 'Q' as the first letter. The remaining letters are E, S, T, U. We need to count the arrangements starting with letters that come before 'U'.

- The next letter in alphabetical order is 'E'. If we fix 'E' as the second letter, the remaining letters are S, T, U. The number of arrangements of these 3 letters is given by \(3!\):
\[ 3! = 6 \]

### Step 4: Fix 'Q' and 'U', and count arrangements starting with letters before 'S'
Now, we fix 'Q' and 'U'. The remaining letters are E, S, T. We need to count the arrangements starting with letters that come before 'S'.

- The next letter in alphabetical order is 'E'. If we fix 'E' as the third letter, the remaining letters are S and T. The number of arrangements of these 2 letters is given by \(2!\):
\[ 2! = 2 \]

### Step 5: Fix 'Q', 'U', and 'E', and count arrangements starting with letters before 'T'
Now, we fix 'Q', 'U', and 'E'. The remaining letters are S and T. We need to count the arrangements starting with letters that come before 'T'.

- The next letter in alphabetical order is 'S'. If we fix 'S' as the fourth letter, the only letter left is 'T'. There is only 1 arrangement:
\[ 1! = 1 \]

### Step 6: Calculate the total number of arrangements before 'QUEST'
Now, we can sum all the arrangements we counted:
- Arrangements starting with 'E': \(24\)
- Arrangements starting with 'QE': \(6\)
- Arrangements starting with 'QU': \(2\)
- Arrangements starting with 'QUE': \(1\)

Total arrangements before "QUEST":
\[ 24 + 6 + 2 + 1 = 33 \]

### Step 7: Find the rank of "QUEST"
The rank of the word "QUEST" is the total arrangements before it plus 1 (for the word itself):
\[ \text{Rank of QUEST} = 33 + 1 = 34 \]

Thus, the rank of the word "QUEST" is **34**.

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