The coefficient of x4 in the expansion of  1+x+x2+x311 is

# The coefficient of x4 in the expansion of  ${\left(1+\mathrm{x}+{\mathrm{x}}^{2}+{\mathrm{x}}^{3}\right)}^{11}$ is

1. A

900

2. B

909

3. C

990

4. D

999

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### Solution:

Let $\mathrm{E}={\left(1+\mathrm{x}+{\mathrm{x}}^{2}+{\mathrm{x}}^{3}\right)}^{11}$

$\begin{array}{r}={\left[\left(1+x\right)\left(1+{x}^{2}\right)\right]}^{11}\\ =\left(1+x{\right)}^{11}{\left(1+{x}^{2}\right)}^{11}\end{array}$

General term of E is,

${=}^{11}{\mathrm{C}}_{\mathrm{r}}{\cdot }^{11}{\mathrm{C}}_{\mathrm{k}}{\mathrm{x}}^{\mathrm{r}+2\mathrm{k}}$

For taking coefficient of x4 take
r+2k=4

. Coefficient of x4

$\begin{array}{l}={\mathrm{T}}_{\left(0,2\right)}+{\mathrm{T}}_{\left(2,1\right)}+{\mathrm{T}}_{\left(4,0\right)}\\ {=}^{11}{{\mathrm{C}}_{0}}^{11}{\mathrm{C}}_{2}{+}^{11}{{\mathrm{C}}_{2}}^{11}{\mathrm{C}}_{1}{+}^{11}{{\mathrm{C}}_{4}}^{11}{\mathrm{C}}_{0}\\ =55+55×11+330\\ =55+605+330=990\end{array}$

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