Question Number 123695 by pipin last updated on 27/Nov/20 $$\boldsymbol{\mathrm{There}}\:\boldsymbol{\mathrm{are}}\:\boldsymbol{\mathrm{how}}\:\boldsymbol{\mathrm{many}}\:\boldsymbol{\mathrm{ways}}\:\boldsymbol{\mathrm{to}}\:\boldsymbol{\mathrm{answer}} \\ $$$$\mathrm{5}\:\boldsymbol{\mathrm{out}}\:\boldsymbol{\mathrm{of}}\:\mathrm{7}\:\boldsymbol{\mathrm{multiple}}\:\boldsymbol{\mathrm{choice}}\:\boldsymbol{\mathrm{question}}\:\boldsymbol{\mathrm{with}} \\ $$$$\mathrm{5}\:\boldsymbol{\mathrm{choices}}? \\ $$ Answered by mr W last updated on 27/Nov/20 $${C}_{\mathrm{5}}…
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Question Number 123577 by bramlexs22 last updated on 26/Nov/20 Answered by Olaf last updated on 26/Nov/20 $$\begin{cases}{\frac{{g}}{{r}+{g}}\:=\:\frac{\mathrm{4}}{\mathrm{9}}\:\left(\mathrm{1}\right)}\\{\frac{{g}+\mathrm{2}}{\left({r}+\mathrm{4}\right)+\left({g}+\mathrm{2}\right)}\:=\:\frac{\mathrm{10}}{\mathrm{23}}\:\left(\mathrm{2}\right)}\end{cases} \\ $$$$\left(\mathrm{1}\right)\::\:{r}+{g}\:=\:\frac{\mathrm{9}}{\mathrm{4}}{g}\:\left(\mathrm{3}\right) \\ $$$$\left(\mathrm{2}\right)\::\:\frac{{g}+\mathrm{2}}{\frac{\mathrm{9}}{\mathrm{4}}{g}+\mathrm{6}}\:\:=\:\frac{\mathrm{10}}{\mathrm{23}} \\ $$$${g}+\mathrm{2}\:=\:\frac{\mathrm{10}}{\mathrm{23}}\left(\frac{\mathrm{9}}{\mathrm{4}}{g}+\mathrm{6}\right) \\ $$$${g}\left(\mathrm{1}−\frac{\mathrm{45}}{\mathrm{46}}\right)\:=\:\frac{\mathrm{60}}{\mathrm{23}}−\mathrm{2}\:=\:\frac{\mathrm{14}}{\mathrm{23}}…
Question Number 57985 by mr W last updated on 15/Apr/19 $${Find}\:{the}\:{number}\:{of}\:{integer}\:{solutions} \\ $$$${for}\:{a}×{b}×{c}×{d}=\mathrm{18900} \\ $$$${with}\:{a},{b},{c},{d}\geqslant\mathrm{1}. \\ $$ Answered by tanmay.chaudhury50@gmail.com last updated on 15/Apr/19 $$\mathrm{18900}=\mathrm{3}×\mathrm{3}×\mathrm{3}×\mathrm{7}×\mathrm{2}×\mathrm{2}×\mathrm{5}×\mathrm{5}…
Question Number 189049 by mr W last updated on 11/Mar/23 Commented by mr W last updated on 13/Mar/23 $${this}\:{is}\:{an}\:{old}\:{question}\:\left({Q}\mathrm{164560}\right).\: \\ $$$${maybe}\:{meanwhile}\:{somebody}\:{has}\:{got}\: \\ $$$${a}\:{new}\:{solution}. \\ $$…
Question Number 189021 by mr W last updated on 10/Mar/23 $${How}\:{many}\:{non}−{similar}\:{triangles} \\ $$$${have}\:{integer}\:{angles}\:{in}\:°? \\ $$ Commented by nikif99 last updated on 11/Mar/23 $${Now}\:{I}\:{think}\:{there}\:{are}\:\mathrm{2700}\:{solutions}\: \\ $$$${for}\:\measuredangle{A},\:\measuredangle{B},\:\measuredangle{C}\:{integers}\:{degrees}.…
Question Number 57909 by mr W last updated on 14/Apr/19 $${n}\:{men}\:{and}\:{n}\:{women}\:{should}\:{be}\:{arranged} \\ $$$${alternately}\:{in}\:{a}\:{row},\:{how}\:{many}\:{ways} \\ $$$${can}\:{this}\:{be}\:{done}?\:{if}\:{the}\:{same}\:{should} \\ $$$${be}\:{done}\:{on}\:{a}\:{table},\:{how}\:{many}\:{ways}\:{then}? \\ $$ Answered by tanmay.chaudhury50@gmail.com last updated on…
Question Number 57688 by Tawa1 last updated on 10/Apr/19 $$\:\:\mathrm{Solve}\:\mathrm{for}\:\:\mathrm{n}:\:\:\:\:\:\:\:\:\underset{\mathrm{i}} {\overset{\mathrm{n}\:−\:\mathrm{1}} {\sum}}\:\:\:\overset{\mathrm{n}} {\:}\mathrm{C}_{\mathrm{i}} \:\mathrm{2}^{\mathrm{i}} \:\:=\:\:\mathrm{65},\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{n}\:\in\:\mathbb{Z}^{+} .\:\:\:\:\mathrm{where}\:\:\mathrm{zero}\:\mathrm{is}\: \\ $$$$\:\:\mathrm{included} \\ $$ Commented by tanmay.chaudhury50@gmail.com last updated…
Question Number 188551 by mr W last updated on 03/Mar/23 $${you}\:{randomly}\:{select}\:{a}\:\mathrm{5}\:{digit}\:{number}. \\ $$$${what}'{s}\:{the}\:{probability}\:{that}\:{this}\:{number} \\ $$$${has}\:{exactly}\:\mathrm{3}\:{different}\:{digits}? \\ $$ Answered by mr W last updated on 03/Mar/23…
Question Number 57363 by Tawa1 last updated on 03/Apr/19 Commented by Tawa1 last updated on 03/Apr/19 $$\mathrm{Please}\:\mathrm{help}.\:\mathrm{my}\:\mathrm{problem}\:\mathrm{is}\:\mathrm{from}\:\:\frac{\mathrm{2}^{\mathrm{m}} \:.\:\mathrm{m}\left(\mathrm{m}\:+\:\mathrm{1}\right)\left(\mathrm{m}\:+\:\mathrm{2}\right)\:….\:\left(\mathrm{2m}\:−\:\mathrm{1}\right)}{\mathrm{1}.\mathrm{3}.\mathrm{5}.\:…….\:\left(\mathrm{2m}\:−\:\mathrm{1}\right)} \\ $$$$\mathrm{now}\:\mathrm{why}\:\mathrm{did}\:\mathrm{they}\:\mathrm{multiply}\:\mathrm{Numerator}\:\mathrm{and}\:\mathrm{denominator}\:\mathrm{by} \\ $$$$\left(\mathrm{m}\:−\:\mathrm{1}\right)!.\:\mathrm{2m}\:\:\mathrm{to}\:\mathrm{get}\:… \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\frac{\mathrm{2}^{\mathrm{m}} \:.\:\mathrm{m}\left(\mathrm{m}\:+\:\mathrm{1}\right)\left(\mathrm{m}\:+\:\mathrm{2}\right)\:….\:\left(\mathrm{2m}\:−\:\mathrm{1}\right)\:×\:\left(\mathrm{m}\:−\:\mathrm{1}\right)!.\:\mathrm{2m}}{\left[\mathrm{1}.\mathrm{3}.\mathrm{5}.\:…….\:\left(\mathrm{2m}\:−\:\mathrm{1}\right)\right]\:…….\:.\:\left(\mathrm{m}\:−\:\mathrm{1}\right)!\:.\:\mathrm{2m}\:}…