Question-173065

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Question Number 173065 by mnjuly1970 last updated on 06/Jul/22

Commented by a.lgnaoui last updated on 06/Jul/22

$$\mathrm{103} \\ $$

Commented by mr W last updated on 06/Jul/22

$$\mathrm{154}\:{solutions}\:? \\ $$

Commented by mnjuly1970 last updated on 06/Jul/22

$$\mathrm{yes}\:\:\mathrm{sir}\:\:\mathrm{W}\:\mathrm{thanks}\:\mathrm{alot}… \\ $$$$\mathrm{grateful}\:…. \\ $$

Answered by mr W last updated on 06/Jul/22

(x_1 +x_2 )^3 +x_3 +x_4 +x_5 =9 with x_i ≥0. say A=(x_1 +x_2 )^3 say B=x_3 +x_4 +x_5 A+B=9 case 1: A=0, B=9 A=0 ⇒x_1 =x_2 =0 ⇒1 possibility B=9: (1+t+t^2 +...)^3 =Σ_(k=0) ^∞ C_2 ^(k+2) t^k ⇒C_2 ^(9+2) =55 solutions case 2: A=1, B=8 A=1 ⇒x_1 =0, x_2 =1 or x_1 =1, x_2 =0 ⇒2 possibilities B=8: ⇒C_2 ^(8+2) =45 possibilities ⇒2×45=90 solutions case 3: A=8, B=1 A=8 ⇒x_1 =0, x_2 =2 or x_1 =x_2 =1 or x_1 =2, x_2 =0 ⇒3 possibilities B=1 ⇒3 possibilities ⇒3×3=9 solutions totally: 55+90+9=154 solutions ✓

$$\left({x}_{\mathrm{1}} +{x}_{\mathrm{2}} \right)^{\mathrm{3}} +{x}_{\mathrm{3}} +{x}_{\mathrm{4}} +{x}_{\mathrm{5}} =\mathrm{9}\:{with}\:{x}_{{i}} \geqslant\mathrm{0}. \\ $$$${say}\:{A}=\left({x}_{\mathrm{1}} +{x}_{\mathrm{2}} \right)^{\mathrm{3}} \\ $$$${say}\:{B}={x}_{\mathrm{3}} +{x}_{\mathrm{4}} +{x}_{\mathrm{5}} \\ $$$${A}+{B}=\mathrm{9} \\ $$$$ \\ $$$${case}\:\mathrm{1}:\:{A}=\mathrm{0},\:{B}=\mathrm{9} \\ $$$${A}=\mathrm{0}\:\Rightarrow{x}_{\mathrm{1}} ={x}_{\mathrm{2}} =\mathrm{0}\:\Rightarrow\mathrm{1}\:{possibility} \\ $$$${B}=\mathrm{9}: \\ $$$$\left(\mathrm{1}+{t}+{t}^{\mathrm{2}} +…\right)^{\mathrm{3}} =\underset{{k}=\mathrm{0}} {\overset{\infty} {\sum}}{C}_{\mathrm{2}} ^{{k}+\mathrm{2}} {t}^{{k}} \\ $$$$\Rightarrow{C}_{\mathrm{2}} ^{\mathrm{9}+\mathrm{2}} =\mathrm{55}\:{solutions} \\ $$$$ \\ $$$${case}\:\mathrm{2}:\:{A}=\mathrm{1},\:{B}=\mathrm{8} \\ $$$${A}=\mathrm{1}\: \\ $$$$\Rightarrow{x}_{\mathrm{1}} =\mathrm{0},\:{x}_{\mathrm{2}} =\mathrm{1}\:{or}\:{x}_{\mathrm{1}} =\mathrm{1},\:{x}_{\mathrm{2}} =\mathrm{0}\:\Rightarrow\mathrm{2}\:{possibilities} \\ $$$${B}=\mathrm{8}: \\ $$$$\Rightarrow{C}_{\mathrm{2}} ^{\mathrm{8}+\mathrm{2}} =\mathrm{45}\:{possibilities} \\ $$$$\Rightarrow\mathrm{2}×\mathrm{45}=\mathrm{90}\:{solutions} \\ $$$$ \\ $$$${case}\:\mathrm{3}:\:{A}=\mathrm{8},\:{B}=\mathrm{1} \\ $$$${A}=\mathrm{8}\: \\ $$$$\Rightarrow{x}_{\mathrm{1}} =\mathrm{0},\:{x}_{\mathrm{2}} =\mathrm{2}\:{or}\:{x}_{\mathrm{1}} ={x}_{\mathrm{2}} =\mathrm{1}\:{or}\:{x}_{\mathrm{1}} =\mathrm{2},\:{x}_{\mathrm{2}} =\mathrm{0}\:\Rightarrow\mathrm{3}\:{possibilities} \\ $$$${B}=\mathrm{1}\:\Rightarrow\mathrm{3}\:{possibilities} \\ $$$$\Rightarrow\mathrm{3}×\mathrm{3}=\mathrm{9}\:{solutions} \\ $$$$ \\ $$$${totally}:\: \\ $$$$\mathrm{55}+\mathrm{90}+\mathrm{9}=\mathrm{154}\:{solutions}\:\checkmark \\ $$

Commented by mnjuly1970 last updated on 06/Jul/22