Q. The coefficient of $x^{4}$ in the expansion of $\left(1+x+x^{2}+x^{3}\right)^{6}$ in powers of $x,$ is_______
Solution:
$\left(1+x+x^{2}+x^{3}\right)^{6}=\left((1+x)\left(1+x^{2}\right)\right)^{6}$
$=(1+x)^{6}\left(1+x^{2}\right)^{6}$
$=\displaystyle\sum_{r=0}^{6}{ }^{6} C_{r} x^{r} \displaystyle\sum_{r=0}^{6}{ }^{6} C_{t} x^{2 t}$
$=\displaystyle\sum_{ r =0}^{6} \displaystyle\sum_{ t =0}^{6}{ }^{6} C _{ r }{ }^{6} C _{ t } x ^{ r +2 t }$
For coefficient of $x^{4} $
$\Rightarrow r+2 t=4$
r
t
0
2
2
1
4
0
Coefficient of $x^4$
$= \,{}^6C_0 \,{}^6C_2 + \,{}^6C_2\,{}^6C_1 + \,{}^6C_4\,{}^6C_0$
$= 120 $
| r | t |
|---|---|
| 0 | 2 |
| 2 | 1 |
| 4 | 0 |