1. Letters in the word

The word is QUEEN.

Its letters are: $Q, U, E, E, N$

There are $5$ letters, with $E$ repeated twice.

In dictionary order, we arrange the distinct letters alphabetically: $E < N < Q < U$

We need to find how many distinct arrangements come before the word QUEEN.

2. Count words before QUEEN by fixing letters one by one

We use the standard lexicographic ranking method.

Step 1: First letter of QUEEN is Q

Available letters: $E,E,N,Q,U$

Letters smaller than $Q$ are $E$ and $N$.

Case 1: First letter is $E$

Remaining letters: $E,N,Q,U$

Number of distinct arrangements: $\frac{4!}{1!}=24$

Case 2: First letter is $N$

Remaining letters: $E,E,Q,U$

Number of distinct arrangements: $\frac{4!}{2!}=12$

So, number of words before those starting with $Q$ is: $24+12=36$

Step 2: Second letter of QUEEN is U

Now fix first letter as $Q$.

Remaining letters: $E,E,N,U$

We want second letter smaller than $U$. Letters smaller than $U$ among remaining are $E$ and $N$.

Case 1: Second letter is $E$

Remaining letters: $E,N,U$

Number of arrangements: $3!=6$

Case 2: Second letter is $N$

Remaining letters: $E,E,U$

Number of arrangements: $\frac{3!}{2!}=3$

So, words before those starting with $QU$ are: $6+3=9$

Running total: $36+9=45$

Step 3: Third letter of QUEEN is E

Now fix first two letters as $QU$.

Remaining letters: $E,E,N$

There is no letter smaller than $E$ among the remaining letters.

So contribution is: $0$

Step 4: Fourth letter of QUEEN is E

Now fix first three letters as $QUE$.

Remaining letters: $E,N$

Again, there is no letter smaller than $E$ among remaining letters.

Contribution: $0$

Step 5: Fifth letter is fixed automatically

No contribution.

3. Rank of QUEEN

Number of words before QUEEN is: $45$

Therefore, position of QUEEN is: $45+1=46$

So the word QUEEN is at the 46th position.

4. Option check

• A: $44$th ❌
• B: $45$th ❌
• C: $46$th ✅
• D: $47$th ❌

Hence, the correct option is C.