Key Concepts and Formulas

• Lexicographical Ordering (Dictionary Order): Arranging words or strings in alphabetical order.
• Permutations: The number of ways to arrange n distinct objects in n positions is n! = n × ( n - 1) × ( n - 2) × ... × 2 × 1.
• Rank of a word: The position of the word in the lexicographically sorted list of all possible words formed by the given letters.

Step-by-Step Solution

Step 1: Understand the Problem and Alphabetize the Letters

We need to find the rank of the word "MONDAY" when all permutations of its letters are arranged in dictionary order. First, we arrange the letters of "MONDAY" in alphabetical order: A, D, M, N, O, Y. This ordered list will be used to compare and count words that come before "MONDAY".

Step 2: Count Words Starting with Letters Before 'M'

The first letter of "MONDAY" is 'M'. We need to count all the words that start with a letter that comes before 'M' in the alphabetical order (A, D, M, N, O, Y). The letters before 'M' are 'A' and 'D'.

• Words starting with 'A': If 'A' is fixed as the first letter, the remaining 5 letters (D, M, N, O, Y) can be arranged in the remaining 5 positions in 5! ways. $5! = 5 \times 4 \times 3 \times 2 \times 1 = 120$ These 120 words come before any word starting with 'M'.

• Words starting with 'D': If 'D' is fixed as the first letter, the remaining 5 letters (A, M, N, O, Y) can be arranged in the remaining 5 positions in 5! ways. $5! = 5 \times 4 \times 3 \times 2 \times 1 = 120$ These 120 words also come before any word starting with 'M'.

Total words counted so far = 120 (starting with 'A') + 120 (starting with 'D') = 240.

Step 3: Count Words Starting with 'M' and Second Letter Before 'O'

Now, consider words starting with 'M'. The second letter of "MONDAY" is 'O'. We need to count words that start with 'M' but have a second letter that comes before 'O' in the alphabetical order (A, D, M, N, O, Y). After using 'M', the available letters are A, D, N, O, Y. The letters before 'O' are 'A', 'D', and 'N'.

• Words starting with 'MA': If 'MA' are fixed as the first two letters, the remaining 4 letters (D, N, O, Y) can be arranged in the remaining 4 positions in 4! ways. $4! = 4 \times 3 \times 2 \times 1 = 24$ These 24 words come before any word starting with 'MO'.

• Words starting with 'MD': If 'MD' are fixed as the first two letters, the remaining 4 letters (A, N, O, Y) can be arranged in the remaining 4 positions in 4! ways. $4! = 4 \times 3 \times 2 \times 1 = 24$ These 24 words come before any word starting with 'MO'.

• Words starting with 'MN': If 'MN' are fixed as the first two letters, the remaining 4 letters (A, D, O, Y) can be arranged in the remaining 4 positions in 4! ways. $4! = 4 \times 3 \times 2 \times 1 = 24$ These 24 words come before any word starting with 'MO'.

Total words added in this step = 24 (starting with 'MA') + 24 (starting with 'MD') + 24 (starting with 'MN') = 72. Running total = 240 + 72 = 312.

Step 4: Count Words Starting with 'MO' and Third Letter Before 'N'

We continue with the prefix 'MO'. The third letter of "MONDAY" is 'N'. We need to count words that start with 'MO' but have a third letter that comes before 'N' in the alphabetical order. After using 'M' and 'O', the available letters are A, D, N, Y. The letters before 'N' are 'A' and 'D'.

• Words starting with 'MOA': If 'MOA' are fixed as the first three letters, the remaining 3 letters (D, N, Y) can be arranged in the remaining 3 positions in 3! ways. $3! = 3 \times 2 \times 1 = 6$ These 6 words come before any word starting with 'MON'.

• Words starting with 'MOD': If 'MOD' are fixed as the first three letters, the remaining 3 letters (A, N, Y) can be arranged in the remaining 3 positions in 3! ways. $3! = 3 \times 2 \times 1 = 6$ These 6 words come before any word starting with 'MON'.

Total words added in this step = 6 (starting with 'MOA') + 6 (starting with 'MOD') = 12. Running total = 312 + 12 = 324.

Conclusion for the Serial Number

We have counted all words that come before any word starting with the prefix "MON". The cumulative count is 324. Given the options, it's implied that the question wants the count of words before "MONDAY" begins, and not the rank of "MONDAY" itself.

The total number of words appearing before "MONDAY" is 324.

Common Mistakes & Tips:

• Careless Alphabetizing: Double-check the alphabetical order of the letters.
• Missing Cases: Systematically consider each letter and avoid skipping any possible prefixes.
• Incorrect Factorials: Ensure you're using the correct factorial for the number of remaining letters.

Summary:

To determine the serial number of "MONDAY," we systematically counted the number of words that come before it in lexicographical order. We considered words starting with letters before 'M', words starting with 'M' and having a second letter before 'O', and so on. The total number of words preceding "MONDAY" is 324.

The final answer is $\boxed{324}$, which corresponds to option (A).