Question Number 151393 by Tawa11 last updated on 20/Aug/21
Answered by mr W last updated on 20/Aug/21
$$\left(\mathrm{1}+{x}+{x}^{\mathrm{2}} +…\right)^{\mathrm{3}} \left(\mathrm{1}+{x}^{\mathrm{5}} +{x}^{\mathrm{10}} +{x}^{\mathrm{15}} +…\right) \\ $$$$=\left(\mathrm{1}+{x}^{\mathrm{5}} +{x}^{\mathrm{10}} +{x}^{\mathrm{15}} +…\right)\underset{{k}=\mathrm{0}} {\overset{\infty} {\sum}}{C}_{\mathrm{2}} ^{{k}+\mathrm{2}} {x}^{{k}} \\ $$$${the}\:{coef}.\:{of}\:{x}^{\mathrm{15}} \:{is} \\ $$$${C}_{\mathrm{2}} ^{\mathrm{15}+\mathrm{2}} +{C}_{\mathrm{2}} ^{\mathrm{10}+\mathrm{2}} +{C}_{\mathrm{2}} ^{\mathrm{5}+\mathrm{2}} +\mathrm{1} \\ $$$$={C}_{\mathrm{2}} ^{\mathrm{17}} +{C}_{\mathrm{2}} ^{\mathrm{12}} +{C}_{\mathrm{2}} ^{\mathrm{7}} +\mathrm{1} \\ $$$$=\mathrm{136}+\mathrm{66}+\mathrm{21}+\mathrm{1} \\ $$$$=\mathrm{224} \\ $$$$\Rightarrow{answer}\:\left(\mathrm{3}\right) \\ $$
Commented by Tawa11 last updated on 20/Aug/21
$$\mathrm{Thanks}\:\mathrm{sir}.\:\mathrm{God}\:\mathrm{bless}\:\mathrm{you} \\ $$
Answered by mr W last updated on 20/Aug/21
$${an}\:{other}\:{not}\:{such}\:{smart}\:{way}. \\ $$$${find}\:{at}\:{first}\:{number}\:{of}\: \\ $$$${non}−{negative}\:{integer}\:{solution}\:{of}\: \\ $$$${x}+{y}+{z}={n}. \\ $$$${x}+{y}={n}−{z} \\ $$$${z}=\mathrm{0}\:\Rightarrow{x}+{y}={n}\:\Rightarrow{n}+\mathrm{1}\:{solutions} \\ $$$${z}=\mathrm{1}\:\Rightarrow{x}+{y}={n}−\mathrm{1}\:\Rightarrow{n}\:{solutions} \\ $$$$… \\ $$$${z}={n}\:\Rightarrow{x}+{y}=\mathrm{0}\:\Rightarrow\:\mathrm{1}\:{solution} \\ $$$${totally}\:\mathrm{1}+\mathrm{2}+\mathrm{3}+…+{n}+\left({n}+\mathrm{1}\right)=\frac{\left({n}+\mathrm{1}\right)\left({n}+\mathrm{2}\right)}{\mathrm{2}}\:{solutions} \\ $$$$ \\ $$$${x}+{y}+{z}+\mathrm{5}{t}=\mathrm{15} \\ $$$${x}+{y}+{z}=\mathrm{15}−\mathrm{5}{t} \\ $$$$\left.\mathrm{1}\right)\:{t}=\mathrm{0}\:\Rightarrow{x}+{y}+{z}=\mathrm{15} \\ $$$$\Rightarrow\frac{\left(\mathrm{15}+\mathrm{1}\right)\left(\mathrm{15}+\mathrm{2}\right)}{\mathrm{2}}=\mathrm{136}\:{solutions} \\ $$$$\left.\mathrm{2}\right)\:{t}=\mathrm{1}\:\Rightarrow{x}+{y}+{z}=\mathrm{10} \\ $$$$\Rightarrow\frac{\left(\mathrm{10}+\mathrm{1}\right)\left(\mathrm{10}+\mathrm{2}\right)}{\mathrm{2}}=\mathrm{66}\:{solutions} \\ $$$$\left.\mathrm{3}\right)\:{t}=\mathrm{2}\:\Rightarrow{x}+{y}+{z}=\mathrm{5} \\ $$$$\Rightarrow\frac{\left(\mathrm{5}+\mathrm{1}\right)\left(\mathrm{5}+\mathrm{2}\right)}{\mathrm{2}}=\mathrm{21}\:{solutions} \\ $$$$\left.\mathrm{4}\right)\:{t}=\mathrm{3}\:\Rightarrow{x}+{y}+{z}=\mathrm{0} \\ $$$$\Rightarrow\mathrm{1}\:{solution} \\ $$$${totally}\:{x}+{y}+{z}+\mathrm{5}{t}=\mathrm{15}\:{has} \\ $$$$\mathrm{136}+\mathrm{66}+\mathrm{21}+\mathrm{1}=\mathrm{224}\:{solutions} \\ $$
Commented by Tawa11 last updated on 20/Aug/21
$$\mathrm{Thanks}\:\mathrm{for}\:\mathrm{your}\:\mathrm{time}\:\mathrm{sir}.\:\mathrm{I}\:\mathrm{really}\:\mathrm{appreciate}.\:\mathrm{God}\:\mathrm{bless}\:\mathrm{you}. \\ $$