In how many ways can two identical red, two identical black, and two identical white balls be placed in cells (one per cell) such that same-colored balls are not adjacent?
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Answer: C. 24
Explanation
Arranging identical balls of three distinct colors requires combinatorial placement methods to satisfy adjacency constraints. Constraints dictate that no two adjacent cells can contain balls sharing the exact same color designation. Rigorous combinatorial enumeration of all valid non adjacent configurations yields exactly twenty four possible arrangements. Combinatorics is the branch of mathematics focused on counting, arrangement, and combination of discrete objects. Permutations with restricted adjacent placements often require recursive counting or case by case tree analysis. Identical items of the same color reduce the total number of distinct arrangements compared to uniquely colored objects.
Color arrangement problems appear frequently in coding theory, pattern generation, and molecular chemistry modeling. Combinatorial logic is essential in computer science algorithms for generating constrained sequences and permutations. Statistical mechanics uses similar combinatorial counting methods to analyze particle distributions and energy states. Fifteen underestimates the total number of valid configurations by missing several permitted combinatorial arrangements. Eighteen fails to account for all possible valid placements that respect the strict non adjacency color rule. Thirty overestimates the total count by including invalid permutations where same colored balls sit adjacently.
- Subject
- Mathematics
- Category
- Mathematics
- Topic
- —
- Difficulty
- medium
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