Question Number 196143 by mr W last updated on 18/Aug/23 $${say}\:{you}\:{have}\:\mathrm{3}\:\left({different}\right)\:{books} \\ $$$${about}\:{mathematics},\:\mathrm{4}\:\left({different}\right) \\ $$$${books}\:{about}\:{physics}\:{and}\:\mathrm{5}\:\left({different}\right) \\ $$$${books}\:{about}\:{chemistry}.\:{in}\:{how}\:{many} \\ $$$${ways}\:{can}\:{you}\:{arrange}\:{them}\:{in}\:{a}\:{shelf} \\ $$$${such}\:{that}\:{no}\:{two}\:{books}\:{from}\:{the}\:{same} \\ $$$${subject}\:{are}\:{adjacent}? \\ $$…
Question Number 195964 by mr W last updated on 15/Aug/23 $${the}\:{family}\:{A}\:{has}\:\mathrm{5}\:{members}\:{and}\:{the} \\ $$$${family}\:{B}\:{has}\:\mathrm{4}\:{members}.\:{there}\:{are}\: \\ $$$$\mathrm{6}\:{personsfrom}\:{other}\:{families}. \\ $$$${in}\:{how}\:{many}\:{ways}\:{can}\:{you}\:{arrange} \\ $$$${these}\:\mathrm{15}\:{persons}\:{around}\:{a}\:{round}\:{table} \\ $$$${such}\:{that}\:{no}\:{member}\:{from}\:{family}\:{A} \\ $$$${and}\:{no}\:{member}\:{from}\:{family}\:{B}\:{are} \\ $$$${next}\:{to}\:{each}\:{other}?…
Question Number 195672 by mr W last updated on 07/Aug/23 $${how}\:{many}\:{different}\:{words}\:{can}\:{be} \\ $$$${formed}\:{from}\:{the}\:{letters}\:{in} \\ $$$$\boldsymbol{{aaacdefgbbbb}} \\ $$$${such}\:{that}\:{a}\:“\boldsymbol{{a}}''\:{and}\:{a}\:“\boldsymbol{{b}}''\:{are}\:{not} \\ $$$${next}\:{to}\:{each}\:{other}? \\ $$$$ \\ $$$$\left({see}\:{also}\:{Q}#\mathrm{195606}\right) \\ $$…
Question Number 195666 by uchihayahia last updated on 07/Aug/23 $$ \\ $$$$\:{sequence}\:{of}\:{string}\:{said}\:{to}\:{be}\:{orderly} \\ $$$$\:{if}\:{element}\:{index}\:{i}\:{different}\:{to}\:{i}+\mathrm{1} \\ $$$$\:{for}\:{example} \\ $$$$\:{aba}\:{has}\:{orderly}\:{value}\:\mathrm{2} \\ $$$$\:{abab}\:{has}\:{orderly}\:{value}\:\mathrm{3} \\ $$$$\:{abaabb}\:{has}\:{orderly}\:{value}\:\mathrm{3} \\ $$$$\:{if}\:{there}\:{are}\:\mathrm{7}\:{a}\:{and}\:\mathrm{13}\:{b} \\…
Question Number 195538 by SLVR last updated on 04/Aug/23 $${Number}\:{of}\:{distributions}\:{of} \\ $$$${n}\:{different}\:{articles}\:{to}\:{r}\:{different}\:\:{boxes} \\ $$$$\left.{so}\:{as}\:\mathrm{1}\right){empty}\:{box}\:{allowed} \\ $$$$\left.\mathrm{2}\right){empty}\:{box}\:{not}\:{allowed} \\ $$$${with}\:{proof}…{kindly}\:{help}\:{me} \\ $$ Commented by SLVR last updated…
Question Number 195191 by Erico last updated on 26/Jul/23 $$\mathrm{Calculer}\underset{\mathrm{x}\rightarrow\mathrm{0}} {\:\mathrm{lim}}\frac{\mathrm{sin}\left(\frac{\mathrm{x}}{\mathrm{2}^{\mathrm{n}} }\right)−\frac{\mathrm{1}}{\mathrm{2}^{\mathrm{n}} }\mathrm{sinx}\:\mathrm{cos}\left(\frac{\mathrm{x}}{\mathrm{2}^{\mathrm{n}} }\right)}{\mathrm{sin}^{\mathrm{2}} \left(\frac{\mathrm{x}}{\mathrm{2}^{\mathrm{n}} }\right)} \\ $$ Terms of Service Privacy Policy Contact: info@tinkutara.com
Question Number 195015 by mustafazaheen last updated on 22/Jul/23 Answered by mathlove last updated on 22/Jul/23 $${ass}\Rightarrow{x}=\mathrm{1} \\ $$$${f}\left(\mathrm{1}\right)=\frac{\mathrm{2}}{\mathrm{0}^{{ln}\mathrm{5}} }=\infty \\ $$$$\mathrm{1}\notin{domf}\left({x}\right) \\ $$ Terms…
Question Number 194960 by Erico last updated on 20/Jul/23 $$\mathrm{Soit}\:{x}>\mathrm{1}.\:\mathrm{On}\:\mathrm{d}\acute {\mathrm{e}finie}\:\mathrm{la}\:\mathrm{suite}\:\left(\mathrm{p}_{\mathrm{n}} \right)\:\mathrm{par}\: \\ $$$$\mathrm{p}_{\mathrm{1}} ={x}\:\:\mathrm{et}\:\forall\mathrm{n}\in\mathrm{IN}^{\ast} \:\:\:\:\:\mathrm{p}_{\mathrm{n}+\mathrm{1}} =\mathrm{2p}_{\mathrm{n}} ^{\mathrm{2}} −\mathrm{1} \\ $$$$\mathrm{Montrer}\:\mathrm{que}\:\underset{\mathrm{n}\rightarrow+\infty} {\mathrm{lim}}\:\underset{\mathrm{k}=\mathrm{1}} {\overset{\mathrm{n}} {\prod}}\left(\mathrm{1}+\frac{\mathrm{1}}{\mathrm{p}_{\mathrm{k}} }\right)=\sqrt{\frac{{x}+\mathrm{1}}{{x}−\mathrm{1}}}…
Question Number 194638 by Erico last updated on 12/Jul/23 $$\mathrm{Prove}\:\mathrm{that}\:\forall{n}\in\mathrm{IN}^{\ast} \:\:\:\:\: \\ $$$$\:\:\:\underset{{k}=\mathrm{1}} {\overset{\mathrm{2}^{{n}} −\mathrm{1}} {\sum}}\:\frac{\mathrm{1}}{{sin}^{\mathrm{2}} \left(\frac{{k}\pi}{\mathrm{2}^{{n}+\mathrm{1}} }\right)}=\:\frac{\mathrm{2}^{\mathrm{2}{n}+\mathrm{1}} −\mathrm{2}}{\mathrm{3}} \\ $$$$\mathrm{Give}\:\mathrm{in}\:\mathrm{terms}\:\mathrm{of}\:{n}\:\:\:\underset{{k}=\mathrm{1}} {\overset{\mathrm{2}^{{n}} −\mathrm{1}} {\sum}}\:\frac{\mathrm{1}}{{sin}^{\mathrm{4}} \left(\frac{{k}\pi}{\mathrm{2}^{{n}+\mathrm{1}}…
Question Number 193864 by pascal889 last updated on 21/Jun/23 Answered by MM42 last updated on 21/Jun/23 $$\frac{\frac{{n}!}{\left({n}−\mathrm{5}\right)!}}{\frac{{n}!}{\mathrm{4}!\left({n}−\mathrm{4}\right)!}}=\mathrm{144}\Rightarrow{n}−\mathrm{4}=\mathrm{6}\Rightarrow{n}=\mathrm{10}\checkmark \\ $$ Terms of Service Privacy Policy Contact:…