The coefficient of x^4 in the expansion of
Question:

The coefficient of $x^{4}$ in the expansion of $\left(1+x+x^{2}+x^{3}\right)^{6}$ in powers of $x$, is_________.

Solution:

$\left(1+x+x^{2}+x^{3}\right)^{6}=\left((1+x)\left(1+x^{2}\right)\right)^{6}$

$=(1+x)^{6}\left(1+x^{2}\right)^{6}$

$=\sum_{r=0}^{6}{ }^{6} \mathrm{C}_{\mathrm{r}} \mathrm{x}^{\mathrm{r}} \sum_{\mathrm{r}=0}^{6}{ }^{6} \mathrm{C}_{\mathrm{t}} \mathrm{x}^{2 \mathrm{t}}$

$=\sum_{r=0}^{6} \sum_{t=0}^{6}{ }^{6} C_{r}{ }^{6} C_{t} x^{r+2 t}$

$=\sum_{r=0}^{6} \sum_{t=0}^{6}{ }^{6} \mathrm{C}_{r}{ }^{6} \mathrm{C}_{1} \mathrm{x}^{r+2 r}$

For coefficient of $x^{4} \Rightarrow r+2 t=4$

Coefficient of $x^{4}$

$={ }^{6} \mathrm{C}_{0}{ }^{6} \mathrm{C}_{2}+{ }^{6} \mathrm{C}_{2}{ }^{6} \mathrm{C}_{1}+{ }^{6} \mathrm{C}_{4}{ }^{6} \mathrm{C}_{0}$

$=120$

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