Question:
Find the coefficient of $x 4$ in the expansion of $\left(1+x+x^{2}+x^{3}\right)^{11}$.
Solution:
Given expression is $\left(1+x+x^{2}+x^{3}\right)^{11}$
$=\left[(1+x)+x^{2}(1+x)\right]^{11}=\left[(1+x)\left(1+x^{2}\right)\right]^{11}=(1+x)^{11} \cdot\left(1+x^{2}\right)^{11}$
$=\left({ }^{11} C_{0}+{ }^{11} C_{1} x+{ }^{11} C_{2} x^{2}+{ }^{11} C_{3} x^{3}+{ }^{11} C_{4} x^{4}+\ldots\right)\left({ }^{11} C_{0}+{ }^{11} C_{1} x^{2}+{ }^{11} C_{2} x^{4}+\ldots\right)$
Coefficient of $x^{4}={ }^{11} C_{0} \times{ }^{11} C_{4}+{ }^{11} C_{1} \times{ }^{11} C_{2}+{ }^{11} C_{2} \times{ }^{11} C_{0}$
$=330+605+55=990$