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There-are-8-Hindi-novels-and-6-English-novels-4-Hindi-novels-and-3-English-novels-are-selected-and-arranged-in-a-row-such-that-they-are-alternate-then-no-of-ways-is-




Question Number 21929 by Tinkutara last updated on 07/Oct/17
There are 8 Hindi novels and 6 English  novels. 4 Hindi novels and 3 English  novels are selected and arranged in a  row such that they are alternate then  no. of ways is
$$\mathrm{There}\:\mathrm{are}\:\mathrm{8}\:\mathrm{Hindi}\:\mathrm{novels}\:\mathrm{and}\:\mathrm{6}\:\mathrm{English} \\ $$$$\mathrm{novels}.\:\mathrm{4}\:\mathrm{Hindi}\:\mathrm{novels}\:\mathrm{and}\:\mathrm{3}\:\mathrm{English} \\ $$$$\mathrm{novels}\:\mathrm{are}\:\mathrm{selected}\:\mathrm{and}\:\mathrm{arranged}\:\mathrm{in}\:\mathrm{a} \\ $$$$\mathrm{row}\:\mathrm{such}\:\mathrm{that}\:\mathrm{they}\:\mathrm{are}\:\mathrm{alternate}\:\mathrm{then} \\ $$$$\mathrm{no}.\:\mathrm{of}\:\mathrm{ways}\:\mathrm{is} \\ $$
Commented by mrW1 last updated on 07/Oct/17
to select 4 from 8 hindi novels there  are C_4 ^8  possibilities    to select 3 from 6 english novels there  are C_3 ^6  possibilities    since the 4 hindi novels and 3 english novels  must alternate in a row, they can only  be placed like this:  HEHEHEH    to arrange the 4 selected hindi novels  there are 4! possibilities    to arrange the 3 selected english novels  there are 3! possibilities    ⇒total ways:  C_4 ^8 ×C_3 ^6 ×4!×3!=201600
$$\mathrm{to}\:\mathrm{select}\:\mathrm{4}\:\mathrm{from}\:\mathrm{8}\:\mathrm{hindi}\:\mathrm{novels}\:\mathrm{there} \\ $$$$\mathrm{are}\:\mathrm{C}_{\mathrm{4}} ^{\mathrm{8}} \:\mathrm{possibilities} \\ $$$$ \\ $$$$\mathrm{to}\:\mathrm{select}\:\mathrm{3}\:\mathrm{from}\:\mathrm{6}\:\mathrm{english}\:\mathrm{novels}\:\mathrm{there} \\ $$$$\mathrm{are}\:\mathrm{C}_{\mathrm{3}} ^{\mathrm{6}} \:\mathrm{possibilities} \\ $$$$ \\ $$$$\mathrm{since}\:\mathrm{the}\:\mathrm{4}\:\mathrm{hindi}\:\mathrm{novels}\:\mathrm{and}\:\mathrm{3}\:\mathrm{english}\:\mathrm{novels} \\ $$$$\mathrm{must}\:\mathrm{alternate}\:\mathrm{in}\:\mathrm{a}\:\mathrm{row},\:\mathrm{they}\:\mathrm{can}\:\mathrm{only} \\ $$$$\mathrm{be}\:\mathrm{placed}\:\mathrm{like}\:\mathrm{this}: \\ $$$$\mathrm{HEHEHEH} \\ $$$$ \\ $$$$\mathrm{to}\:\mathrm{arrange}\:\mathrm{the}\:\mathrm{4}\:\mathrm{selected}\:\mathrm{hindi}\:\mathrm{novels} \\ $$$$\mathrm{there}\:\mathrm{are}\:\mathrm{4}!\:\mathrm{possibilities} \\ $$$$ \\ $$$$\mathrm{to}\:\mathrm{arrange}\:\mathrm{the}\:\mathrm{3}\:\mathrm{selected}\:\mathrm{english}\:\mathrm{novels} \\ $$$$\mathrm{there}\:\mathrm{are}\:\mathrm{3}!\:\mathrm{possibilities} \\ $$$$ \\ $$$$\Rightarrow\mathrm{total}\:\mathrm{ways}: \\ $$$$\mathrm{C}_{\mathrm{4}} ^{\mathrm{8}} ×\mathrm{C}_{\mathrm{3}} ^{\mathrm{6}} ×\mathrm{4}!×\mathrm{3}!=\mathrm{201600} \\ $$
Commented by Tinkutara last updated on 07/Oct/17
Thank you very much Sir!
$$\mathrm{Thank}\:\mathrm{you}\:\mathrm{very}\:\mathrm{much}\:\mathrm{Sir}! \\ $$

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