There are six movie parts numbered from 1 to 6. Find the number of ways in which they be arranged so

Question

There are six movie parts numbered from 1 to 6. Find the number of ways in which they be arranged so that part-1 and part-3 are never together.

Accepted Answer

{"type":"doc","content":[{"content":[{"marks":[{"type":"bold"}],"text":"Step 1 (Total Unrestricted Arrangements):","type":"text"},{"text":" The number of ways to arrange 6 distinct movie parts in a row is:","type":"text"}],"type":"paragraph"},{"attrs":{"latex":"\text{Total arrangements} = 6! = 720"},"type":"math_block"},{"content":[{"text":"Step 2 (Arrangements where Part-1 and Part-3 are together):","type":"text","marks":[{"type":"bold"}]},{"text":" We treat Part-1 and Part-3 as a single "super-part" or block:","type":"text"}],"type":"paragraph"},{"content":[{"type":"list_item","content":[{"content":[{"text":"Total elements to arrange: 5 (the block plus the other 4 parts). The number of arrangements is ","type":"text"},{"attrs":{"latex":"5! = 120"},"type":"math_inline"},{"text":" ways.","type":"text"}],"type":"paragraph"}]},{"content":[{"content":[{"text":"Within the block, Part-1 and Part-3 can be arranged in ","type":"text"},{"attrs":{"latex":"2! = 2"},"type":"math_inline"},{"text":" ways.","type":"text"}],"type":"paragraph"}],"type":"list_item"}],"type":"bullet_list"},{"type":"paragraph","content":[{"text":"Total together arrangements is:","type":"text"}]},{"attrs":{"latex":"\text{Together arrangements} = 5! \times 2! = 120 \times 2 = 240"},"type":"math_block"},{"type":"paragraph","content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step 3 (Never together arrangements):"},{"text":" Subtract the together arrangements from the total arrangements:","type":"text"}]},{"attrs":{"latex":"\text{Valid arrangements} = 720 - 240 = 480"},"type":"math_block"},{"content":[{"marks":[{"type":"bold"}],"text":"Step 4 (Conclusion):","type":"text"},{"text":" The number of ways they can be arranged is exactly ","type":"text"},{"marks":[{"type":"code"}],"text":"480","type":"text"},{"type":"text","text":"."}],"type":"paragraph"}]}

Guide / Hint

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