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Question Number 111690 by bemath last updated on 04/Sep/20
Answered by john santu last updated on 04/Sep/20
$$\:{let}\:\begin{cases}{{white}\:{ball}\:=\:{x}}\\{{red}\:{ball}=\mathrm{2}{x}}\\{{black}\:{ball}=\mathrm{30}−\mathrm{3}{x}}\end{cases} \\ $$$${so}\:{the}\:{number}\:{of}\:{way}\:{to}\:{take}\: \\ $$$$\mathrm{2}\:{white}\:{balls}\:{and}\:\mathrm{1}\:{black}\:{ball}\:{is} \\ $$$${C}_{\mathrm{2}} ^{\:{x}} \:.{C}_{\mathrm{1}} ^{\:\mathrm{30}−\mathrm{3}{x}} \:=\:\mathrm{108}\: \\ $$$$\Leftrightarrow\:\:\frac{{x}^{\mathrm{2}} −{x}}{\mathrm{2}}.\left(\mathrm{30}−\mathrm{3}{x}\right)\:=\:\mathrm{108} \\ $$$$\left({x}^{\mathrm{2}} −{x}\right)\left({x}−\mathrm{10}\right)\:+\:\mathrm{72}\:=\:\mathrm{0} \\ $$$$\Rightarrow\:{x}^{\mathrm{3}} −\mathrm{11}{x}^{\mathrm{2}} +\mathrm{10}{x}+\mathrm{72}\:=\:\mathrm{0} \\ $$$$\Rightarrow\left({x}−\mathrm{4}\right)\left({x}−\mathrm{9}\right)\left({x}+\mathrm{2}\right)\:=\:\mathrm{0} \\ $$$${case}\left(\mathrm{1}\right)\:{for}\:{x}\:=\:\mathrm{4}\:\rightarrow\begin{cases}{{red}\:{ball}\:=\:\mathrm{8}}\\{{black}\:{ball}=\mathrm{18}}\end{cases} \\ $$$${so}\:\bigtriangleup\left({b},{r}\right)\:=\:\mathrm{18}−\mathrm{8}\:=\:\mathrm{10} \\ $$$${case}\left(\mathrm{2}\right)\:{for}\:{x}=\mathrm{9}\:\rightarrow\begin{cases}{{red}\:{ball}=\mathrm{18}}\\{{black}\:{ball}=\mathrm{3}}\end{cases} \\ $$$${so}\:\bigtriangleup\left({b},{r}\right)\:=\:\mathrm{15}. \\ $$$$ \\ $$
Commented by bemath last updated on 04/Sep/20
$${thanks}\:{sir},\:{a}\:{farmer}\:{math} \\ $$
Commented by john santu last updated on 04/Sep/20
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Commented by Rasheed.Sindhi last updated on 04/Sep/20
$${Math}\:{is}\:{a}\:{farm}. \\ $$$${Algebra},{Geometry},{Calculus}, \\ $$$${Trigonometry},{Number}\:{Theory} \\ $$$${etc}\:{are}\:{various}\:{kinds}\:{of}\:{crops} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{AND} \\ $$$$\mathrm{The}\:\mathrm{farmer}\:\mathrm{is}\:\mathrm{Mr}\:\mathrm{John}\:\mathrm{Santu}. \\ $$$$\mathrm{Bemath},\mathrm{I}\:\&\:\mathrm{others}\:\mathrm{are} \\ $$$$\mathrm{consumers}! \\ $$$$\mathrm{Work}\:\mathrm{hard}\:\mathrm{Mr}\:\mathrm{Santu},{because} \\ $$$$\left.{consumption}\:{is}\:{raising}\:{high}!\::\right) \\ $$
Commented by john santu last updated on 04/Sep/20
$${hahaha}...{thank}\:{you}\:{sir}. \\ $$
Commented by bemath last updated on 04/Sep/20
$${agree}\:{sir}\: \\ $$