Method for Finding the Rank of a Word

To find the position (or rank) of the word QUEEN, we need to count how many unique words can be formed that would appear before it in an alphabetical (dictionary) order. The final rank will be this count plus one.

The letters in the word QUEEN are: Q, U, E, E, N.

The alphabetical order of these unique letters is: E, N, Q, U.

Step 1: Words starting with letters before 'Q'

The letters that come before 'Q' in our set are E and N.

• Words starting with E: If the first letter is E, the remaining 4 letters (Q, U, E, N) can be arranged in $4!$ ways.
  Number of words = $4! = 4 \times 3 \times 2 \times 1 = 24$.
• Words starting with N: If the first letter is N, the remaining 4 letters (Q, U, E, E) can be arranged in $\frac{4!}{2!}$ ways (we divide by 2! because 'E' is repeated).
  Number of words = $\frac{24}{2} = 12$.

Step 2: Words starting with 'Q'

The target word QUEEN starts with Q. So, we fix 'Q' as the first letter and look at the second letter. The remaining letters are {U, E, E, N}. In alphabetical order, these are {E, E, N, U}.

The second letter of our word is 'U'. We need to count the words that start with 'Q' but have a second letter that comes before 'U' (i.e., E or N).

• Words starting with QE: If the first two letters are QE, the remaining 3 letters (U, E, N) can be arranged in $3!$ ways.
  Number of words = $3! = 3 \times 2 \times 1 = 6$.
• Words starting with QN: If the first two letters are QN, the remaining 3 letters (U, E, E) can be arranged in $\frac{3!}{2!}$ ways.
  Number of words = $\frac{6}{2} = 3$.

Step 3: Words starting with 'QUE' and beyond

The next word alphabetically will start with 'QU'. The third letter is 'E', which is the first in the remaining set {E, E, N}. The fourth letter is also 'E', which is the first available. The last letter must be 'N'. This means the very next word in the sequence is QUEEN itself.

Step 4: Calculating the Final Rank

The total number of words that come before QUEEN is the sum of the counts from the steps above:

Words before = 24 (starting with E) + 12 (starting with N) + 6 (starting with QE) + 3 (starting with QN) = 45.

The rank of the word QUEEN is this count plus one.

Rank = 45 + 1 = 46.

Conclusion

The position of the word QUEEN is 46th.