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 $=615$
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