Arrangements of CIVILIZATION: How many distinct permutations are there for the letters of “civilization”?
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A12! / 4!
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B12! / 4! -1
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C13! / 5! - 1
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DNone of these
Answer
Correct Answer: 12! / 4!
Explanation
Introduction / Context:The word “civilization” has repeating letters, notably the letter i. Distinct permutations require dividing by factorials of identical letter counts.
Given Data / Assumptions:Total letters = 12; the letter i appears 4 times; assume other letters are distinct for counting purposes.
Concept / Approach:Distinct permutations = 12! / 4! due to the four indistinguishable i’s.
Step-by-Step Solution:
Compute symbolically: 12! / 4!No further identical-letter divisors are needed beyond i^4 for this count.Verification / Alternative check:Letter-by-letter counting confirms four i’s and the remaining letters each occur once in typical textbook treatments.
Why Other Options Are Wrong:Subtracting 1 (as in options b/c) is unjustified; 13!/5! mismatches length and multiplicity.
Common Pitfalls:Miscounting the number of i’s or injecting spurious duplicate counts for unique letters.
Final Answer:12! / 4!