Q96 / 180 · COMEDK 2024 (Memory Based) · Mathematics
The coefficient of $x^6$ in the expansion of $\left(1+x^2+2 x\right)^{10}$ is
- A${ }^{10} C_6$
- B${ }^{10} C_5^6$
- C${ }^{10} C_3$
- Correct answerD${ }^{20} C_6$
Solution
We have, $\left(1+x^2+2 x\right)^{10}=\left[(1+x)^2\right]^0=(1+x)^{20}$
$\therefore$ General term of $(x+y)^n={ }^" C_r(x)^{n-r}(y)^r$
$\Rightarrow \quad T_{r+1}={ }^{20} C_r(1)^{20-r}(x)^r={ }^{20} C_r(x)^r$
To find coefficient of $x^6$ we need to put $r=6$
$\therefore$ Coefficient of $x^6={ }^{20} C_6$
$\therefore$ General term of $(x+y)^n={ }^" C_r(x)^{n-r}(y)^r$
$\Rightarrow \quad T_{r+1}={ }^{20} C_r(1)^{20-r}(x)^r={ }^{20} C_r(x)^r$
To find coefficient of $x^6$ we need to put $r=6$
$\therefore$ Coefficient of $x^6={ }^{20} C_6$
Hence, the correct answer is option: 'D'.