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Question Number 151393 by Tawa11 last updated on 20/Aug/21

Answered by mr W last updated on 20/Aug/21

$${(1+x+x^2+...)}^3(1+x^5+x^{10}+x^{15}+...)$$$$=(1+x^5+x^{10}+x^{15}+...)\underset{k=0}{\sum ^{\mathrm{\infty }}}{C}_2^{k+2}{x}^k$$$$thecoef.of{x}^{15}is$$$$C_2^{15+2}+C_2^{10+2}+C_2^{5+2}+1$$$$=C_2^{17}+C_2^{12}+C_2^7+1$$$$=136+66+21+1$$$$=224$$$$\Rightarrow answer\left(3\right)$$

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

$$anothernotsuchsmartway.$$$$findatfirstnumberof$$$$non-negativeintegersolutionof$$$$x+y+z=n.$$$$x+y=n-z$$$$z=0\Rightarrow x+y=n\Rightarrow n+1solutions$$$$z=1\Rightarrow x+y=n-1\Rightarrow nsolutions$$$$...$$$$z=n\Rightarrow x+y=0\Rightarrow 1solution$$$$totally1+2+3+...+n+(n+1)=\frac{(n+1)(n+2)}{2}solutions$$$$$$$$x+y+z+5t=15$$$$x+y+z=15-5t$$$$1)t=0\Rightarrow x+y+z=15$$$$\Rightarrow \frac{(15+1)(15+2)}{2}=136solutions$$$$2)t=1\Rightarrow x+y+z=10$$$$\Rightarrow \frac{(10+1)(10+2)}{2}=66solutions$$$$3)t=2\Rightarrow x+y+z=5$$$$\Rightarrow \frac{(5+1)(5+2)}{2}=21solutions$$$$4)t=3\Rightarrow x+y+z=0$$$$\Rightarrow 1solution$$$$totallyx+y+z+5t=15has$$$$136+66+21+1=224solutions$$

Commented by Tawa11 last updated on 20/Aug/21

$$\mathrm{Thanks}\mathrm{for}\mathrm{your}\mathrm{time}\mathrm{sir}.\mathrm{I}\mathrm{really}\mathrm{appreciate}.\mathrm{God}\mathrm{bless}\mathrm{you}.$$

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