Question Number 180027 by mr W last updated on 06/Nov/22
$$\mathrm{There}\:\mathrm{are}\:\mathrm{4}\:\mathrm{identical}\:\mathrm{mathematics} \\ $$$$\mathrm{books},\:\mathrm{3}\:\mathrm{identical}\:\mathrm{physics}\:\mathrm{books},\:\mathrm{2} \\ $$$$\mathrm{identical}\:\mathrm{chemistry}\:\mathrm{books}\:\mathrm{and}\:\mathrm{2} \\ $$$$\mathrm{identical}\:\mathrm{biology}\:\mathrm{books}.\:\mathrm{in}\:\mathrm{how}\:\mathrm{many} \\ $$$$\mathrm{ways}\:\:\mathrm{can}\:\mathrm{you}\:\mathrm{compile}\:\mathrm{these}\:\mathrm{books} \\ $$$$\mathrm{such}\:\mathrm{that}\:\mathrm{same}\:\mathrm{books}\:\mathrm{are}\:\mathrm{not}\:\mathrm{mutually} \\ $$$$\mathrm{adjacent}. \\ $$$$\left({an}\:{unsolved}\:{old}\:{question}\right) \\ $$
Answered by cortano1 last updated on 06/Nov/22
$$\:=\:\frac{\mathrm{11}!}{\mathrm{4}!.\mathrm{3}!.\left(\mathrm{2}!\right)^{\mathrm{2}} }\:−\left(\mathrm{4}!\right)^{\mathrm{2}} .\mathrm{3}!.\left(\mathrm{2}!\right)^{\mathrm{2}} \\ $$$$=\mathrm{55476} \\ $$
Commented by cortano1 last updated on 06/Nov/22
$$\:\mathrm{o}\:\mathrm{yes}\:\mathrm{sir}.\:\mathrm{thanks}\:\mathrm{you} \\ $$
Commented by mr W last updated on 06/Nov/22
$$“{same}\:{books}\:{are}\:{not}\:{mutually}\: \\ $$$${adjacent}''\:{means}\:“{no}\:{two}\:{books}\:{from} \\ $$$${the}\:{same}\:{kind}\:{are}\:{next}\:{to}\:{each}\:{other}'', \\ $$$${i}.{e}.\:{left}\:{and}\:{right}\:{to}\:{every}\:\:{book}\:{must}\: \\ $$$${be}\:{books}\:{from}\:{other}\:{kinds}. \\ $$$${you}\:{seam}\:{to}\:{have}\:{understood}\:“{not}\:{all}\: \\ $$$${books}\:{from}\:{the}\:{same}\:{kind}\:{are}\: \\ $$$${together}''. \\ $$
Commented by mr W last updated on 06/Nov/22
$${example}: \\ $$$${MPMPMPMBCBC}\:{is}\:{valid}, \\ $$$${MPMPMMPBCBC}\:{is}\:{not}\:{valid}. \\ $$