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Nine-chairs-in-a-row-are-to-be-occupied-by-six-students-and-Prof-George-Prof-Pieter-and-Prof-John-These-three-professors-arrive-before-the-six-students-and-decide-to-choose-their-chairs-so-that-ea




Question Number 123878 by liberty last updated on 29/Nov/20
Nine chairs in a row are to be occupied by  six students and Prof George , Prof Pieter,  and Prof John. These three professors arrive  before the six students and decide to choose  their chairs so that each professor will be between  two students. In how many ways can Professor George  , Pieter and John choose their chairs ?
$${Nine}\:{chairs}\:{in}\:{a}\:{row}\:{are}\:{to}\:{be}\:{occupied}\:{by} \\ $$$${six}\:{students}\:{and}\:{Prof}\:{George}\:,\:{Prof}\:{Pieter}, \\ $$$${and}\:{Prof}\:{John}.\:{These}\:{three}\:{professors}\:{arrive} \\ $$$${before}\:{the}\:{six}\:{students}\:{and}\:{decide}\:{to}\:{choose} \\ $$$${their}\:{chairs}\:{so}\:{that}\:{each}\:{professor}\:{will}\:{be}\:{between} \\ $$$${two}\:{students}.\:{In}\:{how}\:{many}\:{ways}\:{can}\:{Professor}\:{George} \\ $$$$,\:{Pieter}\:{and}\:{John}\:{choose}\:{their}\:{chairs}\:?\: \\ $$
Answered by john_santu last updated on 29/Nov/20
 let ⊕=place  for 3 professor           △=place for student  △△⊕△△⊕△△⊕△△⊕△△⊕  = P_3 ^( 5)  = ((5!)/(2!)) = 60 ways
$$\:{let}\:\oplus={place}\:\:{for}\:\mathrm{3}\:{professor} \\ $$$$\:\:\:\:\:\:\:\:\:\bigtriangleup={place}\:{for}\:{student} \\ $$$$\bigtriangleup\bigtriangleup\oplus\bigtriangleup\bigtriangleup\oplus\bigtriangleup\bigtriangleup\oplus\bigtriangleup\bigtriangleup\oplus\bigtriangleup\bigtriangleup\oplus \\ $$$$=\:{P}_{\mathrm{3}} ^{\:\mathrm{5}} \:=\:\frac{\mathrm{5}!}{\mathrm{2}!}\:=\:\mathrm{60}\:{ways} \\ $$$$ \\ $$

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