# The coefficient of x{4} in the expansion

Question:

The coefficient of $x^{4}$ in the expansion of $\left(1+x+x^{2}\right)^{10}$ is

Solution:

General term of the expansion $=\frac{10 !}{\alpha ! \beta ! \gamma !} x^{\beta+2 \gamma}$

For coefficient of $x^{4} ; \beta+2 \gamma=4$

Here, three cases arise

Case-1: When $\gamma=0, \beta=4, \alpha=6$

$\Rightarrow \frac{10 !}{6 ! 4 ! 0 !}=210$

Case-2 : When $\gamma=1, \beta=2, \alpha=7$

$\Rightarrow \frac{10 !}{7 ! 2 ! 1 !}=360$

Case-3: When $\gamma=2, \beta=0, \alpha=8$

$\Rightarrow \frac{10 !}{8 ! 0 ! 2 !}=45$

Hence, total $=615$