In-how-many-ways-can-3-men-and-5-women-be-seated-in-a-row-if-3-specific-women-cannot-sit-next-to-each-other-My-solution-arrangements-when-the-3-women-cannot-sit-next-to-eachother-

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Question Number 125191 by Don08q last updated on 08/Dec/20

$$\mathrm{In}\mathrm{how}\mathrm{many}\mathrm{ways}\mathrm{can}3\mathrm{men}\mathrm{and}5$$$$\mathrm{women}\mathrm{be}\mathrm{seated}\mathrm{in}\mathrm{a}\mathrm{row}\mathrm{if}3\mathrm{specific}$$$$\mathrm{women}cannot\mathrm{sit}\mathrm{next}\mathrm{to}\mathrm{each}\mathrm{other}?$$$$$$$$\mathrm{My}\mathrm{solution}:$$$$\mathrm{arrangements}\mathrm{when}\mathrm{the}3\mathrm{women}$$$$cannot\mathrm{sit}\mathrm{next}\mathrm{to}\mathrm{eachother}$$$$=5!\times {}^6{P}_3$$$$=14,400$$$$$$$$\mathrm{Solution}\mathrm{in}\mathrm{book}:$$$$\mathrm{arrangements}\mathrm{with}\mathrm{no}\mathrm{restrictions}=8!$$$$$$$$\mathrm{arrangements}\mathrm{with}\mathrm{the}3\mathrm{women}$$$$\mathrm{seated}\mathrm{next}\mathrm{to}\mathrm{eachother}=3!\times 6!$$$$$$$$\mathrm{arrangements}\mathrm{when}\mathrm{the}3\mathrm{women}$$$$cannot\mathrm{sit}\mathrm{next}\mathrm{to}\mathrm{eachother}$$$$=8!-(3!\times 6!)$$$$=36,000$$$$$$$${\mathrm{I}}^{\prime }\mathrm{m}\mathrm{confused}.\mathrm{Can}\mathrm{I}\mathrm{please}\mathrm{get}\mathrm{some}$$$$\mathrm{explanation}\mathrm{why}\mathrm{my}\mathrm{solution}\mathrm{is}\mathrm{wrong}?$$

Commented by liberty last updated on 09/Dec/20

$$(\bullet )-W_1-W_2-W_3-$$$$treat3menand2remainingwomenas$$$$identical{}^{\prime }{x}^{\prime }.Since{W}_1,W_{2}and{W}_{3}no$$$$adjacent\Rightarrow -W_1\stackrel{x}{-}{W}_2\stackrel{x}{-}{W}_3-$$$$the3remaining{}^{\prime }{x}^{\prime }scanbeputthe4places$$$$arbitrarelyin\left(\begin{array}{c}3+4-1\\ 3\end{array}\right)=\left(\begin{array}{c}6\\ 3\end{array}\right)=20$$$$thenumberofwayarrangementis$$$$givenby20\times 3!\times 5!=120\times 120=14400$$$$$$

Commented by mr W last updated on 09/Dec/20

$$bothofyouarewrongsirs.$$$$youtreatedthecasesthat“notwoof$$$$thethreewomenarenexttoeach$$$${other}^{″}.butthequestionaskesthat$$$$“allthethreewomenarenotnext$$$$toeach{other}^{″}.thatmeansaccording$$$$tothequestionweshouldonlyexclude$$$$thecasesthatallthethreewomen$$$$aretogether.butinyoursolutions$$$$youexcludealsothecasesthattwo$$$$ofthethreewomenaretogether.$$$$thesolutiongiveninbookiscorrect.$$

Commented by mr W last updated on 09/Dec/20

$$again:accordingtoquestionitis$$$$allowedwhentwoofthethreewomen$$$$aretogether.butthisisnotallowed$$$$inyoursolutions.$$

Commented by liberty last updated on 11/Dec/20

$$yeah\dots .imisunderstandingthequestion.$$

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