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?

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:

Verification / 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