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Question Number 180027 by mr W last updated on 06/Nov/22

There are 4 identical mathematics  books, 3 identical physics books, 2  identical chemistry books and 2  identical biology books. in how many  ways  can you compile these books  such that same books are not mutually  adjacent.  (an unsolved old question)

$$\mathrm{There}\:\mathrm{are}\:\mathrm{4}\:\mathrm{identical}\:\mathrm{mathematics} \\ $$$$\mathrm{books},\:\mathrm{3}\:\mathrm{identical}\:\mathrm{physics}\:\mathrm{books},\:\mathrm{2} \\ $$$$\mathrm{identical}\:\mathrm{chemistry}\:\mathrm{books}\:\mathrm{and}\:\mathrm{2} \\ $$$$\mathrm{identical}\:\mathrm{biology}\:\mathrm{books}.\:\mathrm{in}\:\mathrm{how}\:\mathrm{many} \\ $$$$\mathrm{ways}\:\:\mathrm{can}\:\mathrm{you}\:\mathrm{compile}\:\mathrm{these}\:\mathrm{books} \\ $$$$\mathrm{such}\:\mathrm{that}\:\mathrm{same}\:\mathrm{books}\:\mathrm{are}\:\mathrm{not}\:\mathrm{mutually} \\ $$$$\mathrm{adjacent}. \\ $$$$\left({an}\:{unsolved}\:{old}\:{question}\right) \\ $$

Answered by cortano1 last updated on 06/Nov/22

 = ((11!)/(4!.3!.(2!)^2 )) −(4!)^2 .3!.(2!)^2   =55476

$$\:=\:\frac{\mathrm{11}!}{\mathrm{4}!.\mathrm{3}!.\left(\mathrm{2}!\right)^{\mathrm{2}} }\:−\left(\mathrm{4}!\right)^{\mathrm{2}} .\mathrm{3}!.\left(\mathrm{2}!\right)^{\mathrm{2}} \\ $$$$=\mathrm{55476} \\ $$

Commented by cortano1 last updated on 06/Nov/22

 o yes sir. thanks you

$$\:\mathrm{o}\:\mathrm{yes}\:\mathrm{sir}.\:\mathrm{thanks}\:\mathrm{you} \\ $$

Commented by mr W last updated on 06/Nov/22

“same books are not mutually   adjacent” means “no two books from  the same kind are next to each other”,  i.e. left and right to every  book must   be books from other kinds.  you seam to have understood “not all   books from the same kind are   together”.

$$``{same}\:{books}\:{are}\:{not}\:{mutually}\: \\ $$$${adjacent}''\:{means}\:``{no}\:{two}\:{books}\:{from} \\ $$$${the}\:{same}\:{kind}\:{are}\:{next}\:{to}\:{each}\:{other}'', \\ $$$${i}.{e}.\:{left}\:{and}\:{right}\:{to}\:{every}\:\:{book}\:{must}\: \\ $$$${be}\:{books}\:{from}\:{other}\:{kinds}. \\ $$$${you}\:{seam}\:{to}\:{have}\:{understood}\:``{not}\:{all}\: \\ $$$${books}\:{from}\:{the}\:{same}\:{kind}\:{are}\: \\ $$$${together}''. \\ $$

Commented by mr W last updated on 06/Nov/22

example:  MPMPMPMBCBC is valid,  MPMPMMPBCBC is not valid.

$${example}: \\ $$$${MPMPMPMBCBC}\:{is}\:{valid}, \\ $$$${MPMPMMPBCBC}\:{is}\:{not}\:{valid}. \\ $$

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