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Question Number 112934 by mr W last updated on 10/Sep/20

There are 4 identical mathematics books, 2 identical physics books, 2 identical chemistry books and 2 identical biology books. in how many ways can you compile the 10 books such that same books are not mutually adjacent.

$$\mathrm{There}\:\mathrm{are}\:\mathrm{4}\:\mathrm{identical}\:\mathrm{mathematics} \\ $$$$\mathrm{books},\:\mathrm{2}\:\mathrm{identical}\:\mathrm{physics}\:\mathrm{books},\:\mathrm{2} \\ $$$$\mathrm{identical}\:\mathrm{chemistry}\:\mathrm{books}\:\mathrm{and}\:\mathrm{2} \\ $$$$\mathrm{identical}\:\mathrm{biology}\:\mathrm{books}.\:\mathrm{in}\:\mathrm{how}\:\mathrm{many} \\ $$$$\mathrm{ways}\:\:\mathrm{can}\:\mathrm{you}\:\mathrm{compile}\:\mathrm{the}\:\mathrm{10}\:\mathrm{books} \\ $$$$\mathrm{such}\:\mathrm{that}\:\mathrm{same}\:\mathrm{books}\:\mathrm{are}\:\mathrm{not}\:\mathrm{mutually} \\ $$$$\mathrm{adjacent}. \\ $$

Commented by I want to learn more last updated on 11/Sep/20

MMMM PP CC BB 4! × 4! × 2! × 2! × 2! sir i just tried

$$\mathrm{MMMM}\:\mathrm{PP}\:\mathrm{CC}\:\mathrm{BB} \\ $$$$\:\:\:\mathrm{4}!\:\:×\:\mathrm{4}!\:\:×\:\:\mathrm{2}!\:\:×\:\:\mathrm{2}!\:\:×\:\:\mathrm{2}! \\ $$$$\mathrm{sir}\:\mathrm{i}\:\mathrm{just}\:\mathrm{tried} \\ $$

Commented by I want to learn more last updated on 11/Sep/20

Ohh, i did the same book must be together. That is the one am familier with[sir, this one is new to me. I would love to learn it sir

$$\mathrm{Ohh},\:\:\mathrm{i}\:\mathrm{did}\:\mathrm{the}\:\mathrm{same}\:\mathrm{book}\:\mathrm{must}\:\mathrm{be}\:\mathrm{together}. \\ $$$$\mathrm{That}\:\mathrm{is}\:\mathrm{the}\:\mathrm{one}\:\mathrm{am}\:\mathrm{familier}\:\mathrm{with}\left[\mathrm{sir},\:\mathrm{this}\:\mathrm{one}\:\mathrm{is}\:\mathrm{new}\:\mathrm{to}\:\mathrm{me}.\right. \\ $$$$\mathrm{I}\:\mathrm{would}\:\mathrm{love}\:\mathrm{to}\:\mathrm{learn}\:\mathrm{it}\:\mathrm{sir} \\ $$

Commented by I want to learn more last updated on 11/Sep/20

I will wait for it. Thank you sir.

$$\mathrm{I}\:\mathrm{will}\:\mathrm{wait}\:\mathrm{for}\:\mathrm{it}.\:\mathrm{Thank}\:\mathrm{you}\:\mathrm{sir}.\: \\ $$

Commented by mr W last updated on 12/Sep/20

basically the same as Q109387

$${basically}\:{the}\:{same}\:{as}\:{Q}\mathrm{109387} \\ $$

Commented by I want to learn more last updated on 14/Sep/20

Sir, please study the principle of inclusion/exclusion, it will help you. here. I believe you will understand some principle there better than me sir. riomicheal is trying to use it too but he cannot proceed. n(M′∩P′∩C′∩B′)

$$\mathrm{Sir},\:\:\mathrm{please}\:\mathrm{study}\:\mathrm{the}\:\mathrm{principle}\:\mathrm{of}\:\mathrm{inclusion}/\mathrm{exclusion},\:\mathrm{it}\:\mathrm{will}\:\mathrm{help}\:\mathrm{you}. \\ $$$$\mathrm{here}.\:\mathrm{I}\:\mathrm{believe}\:\mathrm{you}\:\mathrm{will}\:\mathrm{understand}\:\mathrm{some}\:\mathrm{principle}\:\mathrm{there}\:\mathrm{better}\:\mathrm{than}\:\mathrm{me}\:\mathrm{sir}. \\ $$$$\mathrm{riomicheal}\:\mathrm{is}\:\mathrm{trying}\:\mathrm{to}\:\mathrm{use}\:\mathrm{it}\:\mathrm{too}\:\mathrm{but}\:\mathrm{he}\:\mathrm{cannot}\:\mathrm{proceed}. \\ $$$$\:\:\mathrm{n}\left(\mathrm{M}'\cap\mathrm{P}'\cap\mathrm{C}'\cap\mathrm{B}'\right) \\ $$

Commented by mr W last updated on 14/Sep/20

the principle is easy to understand. but i don′t think it helps here really.

$${the}\:{principle}\:{is}\:{easy}\:{to}\:{understand}. \\ $$$${but}\:{i}\:{don}'{t}\:{think}\:{it}\:{helps}\:{here}\:{really}. \\ $$

Commented by Tawa11 last updated on 15/Sep/21

nice

$$\mathrm{nice} \\ $$

Answered by Rio Michael last updated on 10/Sep/20

Allow me to present an idea i had ones seen a studied. If we say A is the number of arragments where mathematics books are adjacent, B is the number of arrangments where Physics books are adjacent, C is the number of arrangments where chemistry books are adjacent and D is the number of arrangments where biology books are adjacent. then the answer we need is n (A ′ ∩ B′ ∩C ′ ∩ D′). i will figure out how to get this value.

$$\mathrm{Allow}\:\mathrm{me}\:\mathrm{to}\:\mathrm{present}\:\mathrm{an}\:\mathrm{idea}\:\mathrm{i}\:\mathrm{had}\:\mathrm{ones}\:\mathrm{seen}\:\mathrm{a}\:\mathrm{studied}. \\ $$$$\:\mathrm{If}\:\mathrm{we}\:\mathrm{say}\:{A}\:\mathrm{is}\:\mathrm{the}\:\mathrm{number}\:\mathrm{of}\:\mathrm{arragments}\:\mathrm{where}\:\mathrm{mathematics}\:\mathrm{books} \\ $$$$\mathrm{are}\:\mathrm{adjacent},\:{B}\:\mathrm{is}\:\mathrm{the}\:\mathrm{number}\:\mathrm{of}\:\mathrm{arrangments}\:\mathrm{where}\:\mathrm{Physics}\:\mathrm{books}\:\mathrm{are} \\ $$$$\mathrm{adjacent},\:{C}\:\mathrm{is}\:\mathrm{the}\:\mathrm{number}\:\mathrm{of}\:\mathrm{arrangments}\:\mathrm{where}\:\mathrm{chemistry}\:\mathrm{books}\:\mathrm{are} \\ $$$$\mathrm{adjacent}\:\mathrm{and}\:{D}\:\mathrm{is}\:\mathrm{the}\:\mathrm{number}\:\mathrm{of}\:\mathrm{arrangments}\:\mathrm{where}\:\mathrm{biology}\:\mathrm{books} \\ $$$$\mathrm{are}\:\mathrm{adjacent}.\:\mathrm{then}\:\mathrm{the}\:\mathrm{answer}\:\mathrm{we}\:\mathrm{need}\:\mathrm{is}\: \\ $$$${n}\:\left({A}\:'\:\cap\:{B}'\:\cap{C}\:'\:\cap\:{D}'\right). \\ $$$$\mathrm{i}\:\mathrm{will}\:\mathrm{figure}\:\mathrm{out}\:\mathrm{how}\:\mathrm{to}\:\mathrm{get}\:\mathrm{this}\:\mathrm{value}. \\ $$

Commented by I want to learn more last updated on 11/Sep/20

Am still waiting for your work sir. Thanks

$$\mathrm{Am}\:\mathrm{still}\:\mathrm{waiting}\:\mathrm{for}\:\mathrm{your}\:\mathrm{work}\:\mathrm{sir}.\:\mathrm{Thanks} \\ $$

Commented by Rio Michael last updated on 14/Sep/20

am finding difficulties in computing my answer, been on this for long now

$$\mathrm{am}\:\mathrm{finding}\:\mathrm{difficulties}\:\mathrm{in}\:\mathrm{computing}\: \\ $$$$\mathrm{my}\:\mathrm{answer},\:\mathrm{been}\:\mathrm{on}\:\mathrm{this}\:\mathrm{for}\:\mathrm{long}\:\mathrm{now} \\ $$

Commented by I want to learn more last updated on 14/Sep/20

And me too, that was why i said he should study it, he can understand it better.

$$\mathrm{And}\:\mathrm{me}\:\mathrm{too},\:\mathrm{that}\:\mathrm{was}\:\mathrm{why}\:\mathrm{i}\:\mathrm{said}\:\mathrm{he}\:\mathrm{should}\:\mathrm{study}\:\mathrm{it},\:\mathrm{he}\:\mathrm{can}\:\mathrm{understand} \\ $$$$\mathrm{it}\:\mathrm{better}. \\ $$

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