Key Concepts and Formulas

• Lexicographical Order (Dictionary Order): Arranging words or strings in the order they would appear in a dictionary.
• Permutations: The number of ways to arrange n distinct objects is $n! = n \times (n-1) \times (n-2) \times ... \times 2 \times 1$.
• If we have n distinct objects, the number of permutations of r objects chosen from them is denoted by $P(n, r) = \frac{n!}{(n-r)!}$.

Step-by-Step Solution

Step 1: Arrange the letters of the word PUBLIC in alphabetical order.

The letters of the word PUBLIC are P, U, B, L, I, C. Arranging these in alphabetical order gives us B, C, I, L, P, U. This ordered list will be used to determine the lexicographical order of the words.

Step 2: Count the number of words that come before PUBLIC in the dictionary.

We will count the number of words that start with letters before 'P' in the alphabetical order (B, C, I, L). Then we will proceed to count the number of words starting with 'P' but having the second letter before 'U' and so on.

Step 3: Count words starting with 'B', 'C', 'I', and 'L'.

• Words starting with 'B': The remaining 5 letters (C, I, L, P, U) can be arranged in $5!$ ways. So, the number of words starting with 'B' is $5! = 5 \times 4 \times 3 \times 2 \times 1 = 120$.
• Words starting with 'C': Similarly, the number of words starting with 'C' is $5! = 120$.
• Words starting with 'I': The number of words starting with 'I' is $5! = 120$.
• Words starting with 'L': The number of words starting with 'L' is $5! = 120$.

Step 4: Count words starting with 'P' and having the second letter before 'U'.

The letters alphabetically before 'U' are B, C, I, and L. So we need to check words starting with PB, PC, PI, PL.

• Words starting with 'PB': The remaining 4 letters (C, I, L, U) can be arranged in $4!$ ways. So, the number of words starting with 'PB' is $4! = 4 \times 3 \times 2 \times 1 = 24$.
• Words starting with 'PC': The number of words starting with 'PC' is $4! = 24$.
• Words starting with 'PI': The number of words starting with 'PI' is $4! = 24$.
• Words starting with 'PL': The number of words starting with 'PL' is $4! = 24$.

Step 5: Count words starting with 'PU' and having the third letter before 'B', 'C', 'I', 'L'.

We are looking for words starting with 'PU' and the third letter being B, C, I, L. The next letter in PUBLIC is B.

Step 6: Count words starting with 'PUB' and having the fourth letter before 'L', 'C', 'I'.

We are looking for words starting with 'PUB' and the fourth letter being before L, C, I in the alphabetical order B, C, I, L, P, U. The next letter in PUBLIC is L. The letters before 'L' are 'C', 'I'.

• Words starting with 'PUBC': The remaining 2 letters (I, L) can be arranged in $2!$ ways. So, the number of words starting with 'PUBC' is $2! = 2 \times 1 = 2$.
• Words starting with 'PUBI': The remaining 2 letters (C, L) can be arranged in $2!$ ways. So, the number of words starting with 'PUBI' is $2! = 2 \times 1 = 2$.

Step 7: Count words starting with 'PUBL' and having the fifth letter before 'I', 'C'.

We are looking for words starting with 'PUBL' and the fifth letter being before 'I', 'C' in alphabetical order. The next letter in PUBLIC is 'I'. The letter before 'I' is 'C'.

• Words starting with 'PUBLC': The remaining 1 letter (I) can be arranged in $1!$ ways. So, the number of words starting with 'PUBLC' is $1! = 1$.

Step 8: Count words starting with 'PUBLIC'.

• Words starting with 'PUBLIC': The remaining 0 letters can be arranged in $0!$ way. So, the number of words starting with 'PUBLIC' is $0! = 1$.

Step 9: Calculate the serial number of the word PUBLIC.

The serial number is the number of words that come before PUBLIC plus 1.

Number of words before PUBLIC $= 4 \times 120 + 4 \times 24 + 2 \times 2 + 1 = 480 + 96 + 4 + 1 = 581$. Serial number of PUBLIC $= 581 + 1 = 582$.

The calculation in the original solution appears to have an error: $4 \times 120+4 \times 24+2 \times 2+2 \times 1 = 582$

However, this is incorrect. It should be $4 \times 120+4 \times 24+2 \times 2+1 = 581$. So the rank is 582.

Common Mistakes & Tips

• Carefully arrange the letters in alphabetical order before starting.
• Remember to add 1 to the count of preceding words to get the serial number.
• Double-check the calculations for factorials and sums to avoid arithmetic errors.

Summary

To find the serial number of the word PUBLIC in a dictionary formed by all permutations of its letters, we first arrange the letters in alphabetical order. Then, we count the number of words that precede PUBLIC by considering the letters that come before each letter of PUBLIC in the alphabetical order. Finally, we add 1 to the count to obtain the serial number. The final serial number of the word PUBLIC is 582.

Final Answer

The final answer is \boxed{582}, which corresponds to option (D).