Question Number 209281 by Tawa11 last updated on 06/Jul/24 $$\mathrm{Find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\mathrm{r},\:\mathrm{if}\:\:\overset{\mathrm{10}} {\:}\mathrm{C}_{\mathrm{r}} \:\:=\:\:\overset{\mathrm{10}} {\:}\mathrm{C}_{\mathrm{2r}\:\:+\:\:\mathrm{1}} \\ $$ Commented by klipto last updated on 06/Jul/24 $$\:\:\:\:\:\:\:^{\boldsymbol{\mathrm{n}}} \boldsymbol{\mathrm{C}}_{\mathrm{r}} =^{\mathrm{n}}…
Question Number 208662 by efronzo1 last updated on 20/Jun/24 $$\:\:\frac{\begin{pmatrix}{\mathrm{n}}\\{\mathrm{0}}\end{pmatrix}\:+\mathrm{3}\begin{pmatrix}{\mathrm{n}}\\{\mathrm{1}}\end{pmatrix}\:+\mathrm{5}\begin{pmatrix}{\mathrm{n}}\\{\mathrm{2}}\end{pmatrix}\:+…+\left(\mathrm{2n}+\mathrm{1}\right)\begin{pmatrix}{\mathrm{n}}\\{\mathrm{n}}\end{pmatrix}}{\begin{pmatrix}{\mathrm{n}}\\{\mathrm{1}}\end{pmatrix}\:+\mathrm{2}\begin{pmatrix}{\mathrm{n}}\\{\mathrm{2}}\end{pmatrix}\:+\:\mathrm{3}\begin{pmatrix}{\mathrm{n}}\\{\mathrm{3}}\end{pmatrix}\:+…+\mathrm{n}\begin{pmatrix}{\mathrm{n}}\\{\mathrm{n}}\end{pmatrix}}\:=\frac{\mathrm{23}}{\mathrm{11}} \\ $$$$\:\mathrm{n}=? \\ $$ Answered by Berbere last updated on 20/Jun/24 $${A}=\underset{{k}=\mathrm{0}} {\overset{{n}} {\sum}}\left(\mathrm{2}{k}+\mathrm{1}\right)\begin{pmatrix}{{n}}\\{{k}}\end{pmatrix};\underset{{k}=\mathrm{0}} {\overset{{n}}…
Question Number 208263 by efronzo1 last updated on 09/Jun/24 Answered by Ghisom last updated on 09/Jun/24 $${a}=\mathrm{e}^{\alpha} \:\Rightarrow\:{b}=\mathrm{e}^{\lambda} {a}\wedge{c}=\mathrm{e}^{\mathrm{2}\lambda} {a} \\ $$$${A}=\mathrm{log}_{{c}} \:{a}\:=\frac{\alpha}{\alpha+\mathrm{2}\lambda} \\ $$$${B}=\mathrm{log}_{{b}}…
Question Number 208238 by alcohol last updated on 08/Jun/24 $$\mathrm{S}{how}\:{that} \\ $$$$\frac{\pi}{\mathrm{4}}\:<\:\underset{\mathrm{0}} {\overset{\mathrm{1}} {\int}}\sqrt{\mathrm{1}−{x}^{\mathrm{4}} }{dx}\:{using}\:{x}\:=\:{sint} \\ $$$${show}\:{that}\:\int_{\mathrm{0}} ^{\mathrm{1}} \sqrt{\mathrm{1}−{x}^{\mathrm{4}} }{dx}<\frac{\mathrm{2}\sqrt{\mathrm{2}}}{\mathrm{3}} \\ $$$${using}\:\left(\int_{\mathrm{0}} ^{\mathrm{1}} {f}\left({x}\right){g}\left({x}\right){dx}\right)^{\mathrm{2}} <\int_{\mathrm{0}}…
Question Number 207937 by efronzo1 last updated on 31/May/24 $$\:\:\mathrm{An}\:\mathrm{ordered}\:\mathrm{data}\:\mathrm{consists}\:\mathrm{of}\: \\ $$$$\:\mathrm{6}\:\mathrm{even}\:\mathrm{numbers}\:\mathrm{and}\:\mathrm{4}\:\mathrm{odd}\:\mathrm{numbers}. \\ $$$$\:\mathrm{The}\:\mathrm{average}\:\mathrm{of}\:\mathrm{the}\:\mathrm{odd}\:\mathrm{numbers}\: \\ $$$$\:\mathrm{is}\:\mathrm{2022}.\:\mathrm{The}\:\mathrm{3rd},\:\mathrm{5th},\:\mathrm{6th}\:\mathrm{and} \\ $$$$\:\mathrm{8}\:\mathrm{th}\:\mathrm{numbers}\:\mathrm{are}\:\mathrm{odd}.\: \\ $$$$\:\mathrm{The}\:\mathrm{data}\:\mathrm{range}\:\mathrm{is}\:\mathrm{24}\:,\:\mathrm{and}\:\mathrm{the}\: \\ $$$$\:\mathrm{interquartile}\:\mathrm{range}\:\mathrm{is}\:\mathrm{14}.\: \\ $$$$\:\mathrm{The}\:\mathrm{largest}\:\mathrm{possible}\:\mathrm{average} \\…
Question Number 207179 by efronzo1 last updated on 08/May/24 Answered by mr W last updated on 08/May/24 $$\mathrm{3}×\mathrm{4}×\mathrm{3}×\mathrm{2}=\mathrm{72}\:{such}\:{numbers} \\ $$ Commented by efronzo1 last updated…
Question Number 206827 by Fabricista15 last updated on 27/Apr/24 $$\mathrm{log}\:\left({x}\right)\:=\:\mathrm{sin}\:\left({x}\right) \\ $$$${x}\:=\:? \\ $$ Commented by Ghisom last updated on 27/Apr/24 $$\mathrm{if}\:\mathrm{log}\:{x}\:=\mathrm{ln}\:{x}\:\Rightarrow \\ $$$$\:\:\:\:\:{x}\approx\mathrm{2}.\mathrm{21910715} \\…
Question Number 206681 by cortano21 last updated on 22/Apr/24 $$\:\:\:\cancel{{s}} \\ $$$$ \\ $$ Commented by A5T last updated on 22/Apr/24 $${This}\:{question}\:{is}\:{similar}\:{to}\:{Q}\mathrm{202257},\:{does}\:{it} \\ $$$${also}\:{mean}\:{every}\:{wife}\:{cannot}\:{come}\:{before}\:{her} \\…
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Question Number 206156 by cortano21 last updated on 08/Apr/24 Commented by mr W last updated on 08/Apr/24 $${what}\:{i}\:{think}\:{is}\:{clear}: \\ $$$${in}\:{the}\:{package}\:{there}\:{must}\:{be}\:{exactly} \\ $$$${one}\:{egg}.\:{there}\:{must}\:{be}\:{at}\:{least}\:{three} \\ $$$${lettuces}.\:{there}\:{must}\:{be}\:{at}\:{least}\:{one} \\…