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Question-173065




Question Number 173065 by mnjuly1970 last updated on 06/Jul/22
Commented by a.lgnaoui last updated on 06/Jul/22
103
$$\mathrm{103} \\ $$
Commented by mr W last updated on 06/Jul/22
154 solutions ?
$$\mathrm{154}\:{solutions}\:? \\ $$
Commented by mnjuly1970 last updated on 06/Jul/22
yes  sir  W thanks alot...  grateful ....
$$\mathrm{yes}\:\:\mathrm{sir}\:\:\mathrm{W}\:\mathrm{thanks}\:\mathrm{alot}… \\ $$$$\mathrm{grateful}\:…. \\ $$
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
     excellent  as always sir W..
$$\:\:\:\:\:\mathrm{excellent}\:\:\mathrm{as}\:\mathrm{always}\:\mathrm{sir}\:\mathrm{W}.. \\ $$
Commented by mr W last updated on 06/Jul/22
thanks sir!
$${thanks}\:{sir}! \\ $$
Commented by Tawa11 last updated on 06/Jul/22
Great sir
$$\mathrm{Great}\:\mathrm{sir} \\ $$

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