The coefficient of $x^7$ in ${(1 - x + 2{x^3})^{10}}$ is ___________.
Answer (integer)
960
Solution
Given expression is $\left(1-x+2 x^3\right)^{10}$
<br/><br/>So, general term is $\frac{10 !}{r_{1} ! r_{2} ! r_{3} !}(1)^{r_1}(-1)^{r_2} \cdot(2)^{r_3} \cdot(x)^{r_2+r_3}$
<br/><br/>Where, $r_1+r_2+r_3=10$ and $r_2+3 r_3=7$
<br/><br/>Now, for possibility,
<br/><br/>$\begin{array}{ccc}r_1 & r_2 & r_3 \\ 3 & 7 & 0 \\ 7 & 1 & 2 \\ 5 & 4 & 1\end{array}$
<br/><br/>Thus, required co-efficient
<br/><br/>$$
\begin{aligned}
& =\frac{10 !}{3 ! 7 !}(-1)^7+\frac{10 !}{5 ! 4 !}(-1)^4(2)+\frac{10 !}{7 ! 2 !}(-1)^1(2)^2 \\\\
& =-120+2520-1440 \\\\
& =2520-1560=960
\end{aligned}
$$
About this question
Subject: Mathematics · Chapter: Binomial Theorem · Topic: Binomial Expansion
This question is part of PrepWiser's free JEE Main question bank. 193 more solved questions on Binomial Theorem are available — start with the harder ones if your accuracy is >70%.