Key Concepts and Formulas

• Permutations: The number of ways to arrange $n$ distinct objects is $n! = n \times (n-1) \times (n-2) \times \dots \times 2 \times 1$.
• Permutations with Repetition: The number of ways to arrange $n$ objects where there are $n_1$ identical objects of type 1, $n_2$ identical objects of type 2, ..., $n_k$ identical objects of type k is $\frac{n!}{n_1! n_2! \dots n_k!}$.
• Lexicographical Order: This refers to the order in which words appear in a dictionary.

Step-by-Step Solution

Step 1: List the letters of the word QUEEN in alphabetical order. The letters are E, E, N, Q, U. We will arrange the words in dictionary order (lexicographical order).

Step 2: Find the number of words starting with E. If the word starts with E, the remaining 4 letters (E, N, Q, U) can be arranged in 4! ways. Therefore, the number of words starting with E is $4! = 4 \times 3 \times 2 \times 1 = 24$.

Step 3: Find the number of words starting with N. If the word starts with N, the remaining 4 letters (E, E, Q, U) can be arranged in $\frac{4!}{2!}$ ways because there are two identical E's. Therefore, the number of words starting with N is $\frac{4!}{2!} = \frac{4 \times 3 \times 2 \times 1}{2 \times 1} = 12$.

Step 4: Find the number of words starting with QE. If the word starts with QE, the remaining 3 letters (E, N, U) can be arranged in 3! ways. Therefore, the number of words starting with QE is $3! = 3 \times 2 \times 1 = 6$.

Step 5: Find the number of words starting with QN. If the word starts with QN, the remaining 3 letters (E, E, U) can be arranged in $\frac{3!}{2!}$ ways because there are two identical E's. Therefore, the number of words starting with QN is $\frac{3!}{2!} = \frac{3 \times 2 \times 1}{2 \times 1} = 3$.

Step 6: Find the number of words starting with QUE. If the word starts with QUE, the remaining 2 letters (E, N) can be arranged in 2! ways. The words are QUEN and QUEN. Since we want to find the position of QUEEN, we need to consider the word QUEN. Therefore, the number of words starting with QUE that come before QUEEN is just the word QUEN, so there is 1 such word.

Step 7: Calculate the position of the word QUEEN. The position of the word QUEEN is the sum of the number of words starting with E, N, QE, QN, and the number of words starting with QUE that come before QUEEN, plus 1 (for the word QUEEN itself). Position of QUEEN = (Number of words starting with E) + (Number of words starting with N) + (Number of words starting with QE) + (Number of words starting with QN) + (Number of words starting with QUEN) + 1 Position of QUEEN = $24 + 12 + 6 + 3 + 1 + 1 = 47$.

The position of the word QUEEN is 46. There appears to be an error in step 6. Let's rethink this step. We've counted words starting with E, N, QE, QN. The next possible words start with QU. Then the possible arrangements are QUE, QUN. We need to find where QUEEN falls. Words starting with QUE: QUEEN Words starting with QUN: QUNE The words before QUEEN are: words starting with E, N, QE, QN. So the number of words before QUEEN are 24 + 12 + 6 + 3 = 45. Then QUEEN is the 46th word.

Let's reconsider step 6 and 7. Words starting with QUE: The remaining letters are E, N. These can be arranged as EN, NE. So we have QUEN, QUENE Words starting with QN: The remaining letters are E, E, U. These can be arranged as EEU, EUE, UEE. So we have QNEE U, QNEUE, QNUEE So the words before QUEEN are words starting with E, N, QE, QN, and QUEN So it is 24 + 12 + 6 + 3 + 0 = 45, then QUEEN is at the position 45 + 1 = 46.

The mistake in the original solution is in the final step. The word QUEEN is the next word after the sum. Therefore, the position of the word QUEEN is 24 + 12 + 6 + 3 + 1 = 46

Common Mistakes & Tips

• Be careful to account for repeated letters when calculating permutations. Divide by the factorial of the number of repetitions.
• Remember to add 1 to the sum of the counts to find the actual position of the word.
• Double-check your arithmetic to avoid errors in the final calculation.

Summary

To find the position of the word QUEEN in the lexicographical order of all possible arrangements of its letters, we count the number of words that come before it. We count the number of words starting with E, N, QE, and QN. Adding these counts gives us the number of words that appear before QUEEN in the dictionary order. Adding 1 to this sum gives the position of the word QUEEN. Therefore, the position of the word QUEEN is $24 + 12 + 6 + 3 + 0 + 1 = 46$.

Final Answer

The final answer is \boxed{46}, which corresponds to option (C).