The coefficient of $x^{7}$ in $\left(1-x+2 x^{3}\right)^{10}$ is

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Professor PC Thomas Classes · Answer

(960) : General term is given by\ $\frac{10!}{r_{1}!r_{2}!r_{3}!}(1)^{r_{2}}(-x)^{r_{2}}\left(2 x^{3}ight)^{r_{3}}$ i.e.,\ $\frac{10!}{r_{1}!\cdot r_{2}!\cdot r_{3}!}(-1)^{r_{2}} \cdot(2)^{r_{3}} x^{r_{2}+3 r_{3}}$\ where $r_{1}+r_{2}+r_{3}=10$ and $r_{2}+3 r_{3}=7$\ Possible values of $\left(r_{1}, r_{2}, r_{3}ight)$, are $(3,7,0)$, $(5,4,1)$ and $(7,1,2)$.\ Required coefficient\ $=\frac{10!}{3!7!}(-1)^{7}+\frac{10!}{5!4!}(-1)^{4}(2)+\frac{10!}{7!2!}(-1)^{1}(2)^{2}$\ $=-120+2520-1440=960$\