Linear arrangements avoiding adjacency of two specified people: Six distinct players stand in a line. In how many ways can they be arranged if Abhinav and Manjesh are never together?
Aptitude
Permutation and Combination
Difficulty: Medium
Choose an option
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A120
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B240
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C360
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D480
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ENone of these
Answer
Correct Answer: 480
Explanation
Introduction / Context:To count arrangements where two specified people are not adjacent, use total permutations minus the arrangements where they are together (treated as a block) times their internal swaps.
Given Data / Assumptions:
- Six distinct players; two are special (A, M).
- We consider linear orderings.
- Prohibited: A and M adjacent.
Concept / Approach:
- Total linear orders = 6!.
- Adjacent count: treat (A,M) as a block with 2 internal orders.
- Not-adjacent = total − adjacent.
Step-by-Step Solution:
Total = 6! = 720Adjacent: block + 4 others → 5! arrangements, and (A,M) can be (A,M) or (M,A) → 2!, so 5!*2 = 120*2 = 240Not-adjacent = 720 − 240 = 480Verification / Alternative check:Gap method (place 4 others, count gaps for A and M) yields the same 480 result, confirming correctness.
Why Other Options Are Wrong:
- 120 and 240 correspond to partial counts (e.g., adjacent cases only).
- 360 still undercounts the exclusions.
Common Pitfalls:
- Forgetting the 2 internal arrangements of the adjacent pair.
Final Answer:480