There-are-4-identical-mathematics-books-2-identic-physics-books-and-2-identical-chemistry-books-How-many-ways-to-compile-the-eight-books-on-the-condition-of-the-same-book-are-not-mutually-adjacent

Question Number 101693 by bemath last updated on 04/Jul/20

$$\mathrm{There}\mathrm{are}4\mathrm{identical}\mathrm{mathematics}$$$$\mathrm{books},2\mathrm{identic}\mathrm{physics}\mathrm{books}$$$$\mathrm{and}2\mathrm{identical}\mathrm{chemistry}\mathrm{books}$$$$.\mathrm{How}\mathrm{many}\mathrm{ways}\mathrm{to}\mathrm{compile}$$$$\mathrm{the}\mathrm{eight}\mathrm{books}\mathrm{on}\mathrm{the}\mathrm{condition}$$$$\mathrm{of}\mathrm{the}\mathrm{same}\mathrm{book}\mathrm{are}\mathrm{not}\mathrm{mutually}$$$$\mathrm{adjacent}?$$

Commented by bobhans last updated on 04/Jul/20

MPMPMCMC, MPMCMPMC, MPMCMCMP, MCMPMPMC, MCMPMCMP, MCMCMPMP, PMPMCMCM, PMCMPMCM, PMCMCMPM, CMPMPMCM, CMPMCMPM, CMCMPMPM, MPMCMPCM, MPMCMCPM, MCMPMPCM, MCMPMCPM, MPMPCMCM, MPMCPMCM, MCMPCMPM, MCMCPMPM, MPCMPMCM, MCPMPMCM, MPCMCMPM, MCPMCMPM

Commented by bemath last updated on 04/Jul/20

$$\mathrm{what}\mathrm{the}\mathrm{formula}?$$

Answered by bobhans last updated on 04/Jul/20

$$24\mathrm{ways}$$

Commented by bobhans last updated on 04/Jul/20

$$\mathrm{M}\mathrm{\_}\mathrm{M}\mathrm{\_}\mathrm{M}\mathrm{\_}\mathrm{M}\mathrm{\_}\iff {\mathrm{P}}_4^4=\frac{4!}{(4-4)!}=24★$$

Commented by mr W last updated on 04/Jul/20

$$canyougivesomeexplanationto$$$$theformulasir?$$

Answered by mr W last updated on 04/Jul/20

$$case1:XMXMXMXM$$$$\Rightarrow \frac{4!}{2!2!}=6$$$$case2:MXMXMXMX$$$$\Rightarrow \frac{4!}{2!2!}=6$$$$case3:MXYMXMXM,MXMXYMXM,MXMXMXYM$$$$\Rightarrow 3\times 2\times 2=12$$$$$$$$totally:6+6+12=24$$

Answered by john santu last updated on 05/Jul/20

$$Ifthemathbooksareinposition$$$$1/3/5/7or2/4/6/8thereare$$$$6waystoarrangethephysics$$$$andthechemistry\to 2\times 6=12$$$$butifthemathbooksarein$$$$position1/3/5/8or1/3/6/8$$$$or1/4/6/8thenthereareonly$$$$4way\to 3\times 4=12$$$$sototally=24ways(JS⊛)$$

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