The coefficient of x^7 in (1 - x + 2x^3)^10 is ___________.

Question

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

Answer: 960

Accepted Answer

Given expression is ${(1-x+2{x}^{3})}^{10}$

So, general term is $\frac{10!}{{r}_{1}!{r}_{2}!{r}_{3}!}(1{)}^{{r}_{1}}(-1{)}^{{r}_{2}}\cdot (2{)}^{{r}_{3}}\cdot (x{)}^{{r}_{2}+{r}_{3}}$

Where, ${r}_{1}+{r}_{2}+{r}_{3}=10$ and ${r}_{2}+3{r}_{3}=7$

Now, for possibility,

$\begin{gathered} {r}_{1} \quad  {r}_{2} \quad  {r}_{3} \ 3 \quad  7 \quad  0 \ 7 \quad  1 \quad  2 \ 5 \quad  4 \quad  1 \end{gathered}$

Thus, required co-efficient

$\begin{gathered} =\frac{10!}{3!7!}(-1{)}^{7}+\frac{10!}{5!4!}(-1{)}^{4}(2)+\frac{10!}{7!2!}(-1{)}^{1}(2{)}^{2} \ =-120+2520-1440 \ =2520-1560=960 \end{gathered}$

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