Arrangements of CIVILIZATION: How many distinct permutations are there for the letters of “civilization”?

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:

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!