If the letters of the word CHASM are rearranged in all possible different 5 letter words and these words are listed in dictionary (lexicographic) order, what is the rank (position) of the word CHASM in this ordered list?

Introduction / Context:
This question is about finding the lexicographic rank of a word formed from distinct letters. You must imagine all permutations of the letters arranged in dictionary order and determine the position of a given word in that list. This is a typical application of permutations with positional counting.

Given Data / Assumptions:

• The word is CHASM.
• Letters are C, H, A, S, M, all distinct.
• All permutations of these 5 letters are considered.
• The words are sorted alphabetically as in a dictionary.
• We want the rank of CHASM among these permutations.

Concept / Approach:
We calculate how many words would appear before CHASM when all permutations are listed in dictionary order. For each position from left to right, we count permutations that start with a smaller letter than the current one, given the remaining letters. The final rank is one plus the total number of words that would appear before the target word.

Step-by-Step Solution:
Step 1: List the letters of CHASM in alphabetical order: A, C, H, M, S.Step 2: Consider the first position. The first letter of CHASM is C. Letters smaller than C in the sorted list are only A.Step 3: For all words starting with A, the remaining 4 letters can be arranged in 4! ways, which is 24. These 24 words come before any word starting with C.Step 4: Now fix the first letter as C. Remaining letters are H, A, S, M. Sort them: A, H, M, S. The second letter in CHASM is H.Step 5: Letters smaller than H among the remaining ones are only A. If second letter is A, the remaining 3 letters can be arranged in 3! = 6 ways. So 6 more words come before CHASM.Step 6: Now fix the first two letters as C and H. Remaining letters are A, S, M. The third letter in CHASM is A. There is no letter smaller than A among these, so add 0.Step 7: Fix CHA and remaining letters S and M. Sorted, they are M, S. The fourth letter in CHASM is S. Among M and S, the letter smaller than S is M, giving 1 additional word CHAM followed by the remaining letter.Step 8: Sum words before CHASM: 24 + 6 + 0 + 1 = 31. Therefore, CHASM is the 32nd word.

Verification / Alternative check:

Why Other Options Are Wrong:

• 24: This corresponds to the total number of words beginning with A, but ignores words starting with C A that precede CHASM.
• 36 and 72: These overestimate the rank and suggest incorrect counting, perhaps forgetting to sort letters correctly or miscomputing factorial contributions at each step.

Common Pitfalls:
Students often forget to sort the letters first before counting words starting with smaller first letters. Another frequent mistake is to ignore the influence of subsequent positions, or to use combinations instead of permutations. Careful tracking of remaining letters and using factorial counts at each position is crucial for accurate ranking.

Final Answer:
The rank of the word CHASM in dictionary order is 32.