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

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Solution

$(1 + x + x^{2} + x^{3})^{11}$
$((1 + x) (1 + x^{2}))^{11}$
$(1 + x)^{11} (1 + x^{2})^{11}$
$[11_{C_{0}} (x^{0})$ $+ 11_{C_{1}} (x^{1})$ $+ 11_{C_{2}} (x^{2})$ $+ . . . . . + 11_{C_{1} 1} (x^{11})]$ $[11_{C_{0}} (x^{2})^{0}$ $+ 11_{C_{1}} (x^{2})^{1}$ $+ 11_{C_{2}} (x^{2})^{2}$ $+ . . . . . + 11_{C_{1} 1} (x^{2})^{11}]$
Coefficient of $x^{4}$ is
$11_{C_{2}} {.11}_{C_{0}}$ $+ 11_{C_{1}} {.11}_{C_{2}}$ $+ 11_{C_{0}} {.11}_{C_{4}}$
$= \frac{11!}{9! 2!} .1$ $+ \frac{11!}{10! 1!} .$ $\frac{11!}{9! 2!}$ $+ \frac{11!}{4! 7!} .1$
$= 11 \times 5 + 11 \times 11 \times 5 + 110 \times 3 = 55 + 605 + 330 = 990$

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