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Question Number 113710 by mr W last updated on 14/Sep/20

you have 2 identical mathematics  books, 2 identical physics books, 2  identical chemistry books, 2 identical  biology books and 2 geography books.  in how many ways can you compile  these books such that same books  are not mutually adjacent?

$${you}\:{have}\:\mathrm{2}\:{identical}\:{mathematics} \\ $$$${books},\:\mathrm{2}\:{identical}\:{physics}\:{books},\:\mathrm{2} \\ $$$${identical}\:{chemistry}\:{books},\:\mathrm{2}\:{identical} \\ $$$${biology}\:{books}\:{and}\:\mathrm{2}\:{geography}\:{books}. \\ $$$${in}\:{how}\:{many}\:{ways}\:{can}\:{you}\:{compile} \\ $$$${these}\:{books}\:{such}\:{that}\:{same}\:{books} \\ $$$${are}\:{not}\:{mutually}\:{adjacent}? \\ $$

Commented by I want to learn more last updated on 16/Sep/20

Great sir,  But does this method solve your previous question?  Q112934

$$\mathrm{Great}\:\mathrm{sir}, \\ $$$$\mathrm{But}\:\mathrm{does}\:\mathrm{this}\:\mathrm{method}\:\mathrm{solve}\:\mathrm{your}\:\mathrm{previous}\:\mathrm{question}? \\ $$$$\mathrm{Q112934} \\ $$

Commented by mr W last updated on 16/Sep/20

no

$${no} \\ $$

Answered by mr W last updated on 15/Sep/20

((10!)/2^5 )−C_1 ^5 ×((9!)/2^4 )+C_2 ^5 ×((8!)/2^3 )−C_3 ^5 ×((7!)/2^2 )+C_4 ^5 ×((6!)/2)−5!  =39480

$$\frac{\mathrm{10}!}{\mathrm{2}^{\mathrm{5}} }−{C}_{\mathrm{1}} ^{\mathrm{5}} ×\frac{\mathrm{9}!}{\mathrm{2}^{\mathrm{4}} }+{C}_{\mathrm{2}} ^{\mathrm{5}} ×\frac{\mathrm{8}!}{\mathrm{2}^{\mathrm{3}} }−{C}_{\mathrm{3}} ^{\mathrm{5}} ×\frac{\mathrm{7}!}{\mathrm{2}^{\mathrm{2}} }+{C}_{\mathrm{4}} ^{\mathrm{5}} ×\frac{\mathrm{6}!}{\mathrm{2}}−\mathrm{5}! \\ $$$$=\mathrm{39480} \\ $$

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