# Find the coefficient of $x^4$ in the expansion of $(1+x+x^2+x^3)^{11}$

$\begin{array}{1 1}(A)\;1000\\(B)\;990\\(C)\;900\\(D)\;800\end{array}$

Toolbox:
• $T_{r+1}=nC_r a^{n-r} b^r$
$(1+x+x^2+x^3)^{11}=\big[\large\frac{1.(1-x^4)}{1-x}\big]^{11}$
$\Rightarrow (1-x^4)^{11}(1-x)^{-11}$
$\Rightarrow (1+11C_1(-x)^4+...)\times (1+(-11)(-x)+\large\frac{(-11)(-12)}{1.2}$$(-x^2) Coefficient of x^4=\large\frac{(-11)(-12)(-13)(-14)}{1.2.3.4}$$-11C_1$
$\Rightarrow 1001-11$
$\Rightarrow 990$
Hence (B) is the correct answer.