QJEEM23Ma3cd2: JEE Main 2023 Mathematics - Class 11 JEEM23Ma3cd2 Solution: The coefficient of in is ___________.

Question

Verified Solution • 2025–26 Session moderate

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

Accepted Answer

Given expression is

\left(1-x+2 x^3\right)^{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{array}{ccc}r_1 &amp; r_2 &amp; r_3 \\ 3 &amp; 7 &amp; 0 \\ 7 &amp; 1 &amp; 2 \\ 5 &amp; 4 &amp; 1\end{array}

Thus, required co-efficient

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

Step-by-Step Verified Solution