The coefficient of $x^{4}$ in the expansion of $(1 + x + x^{2} + x^{3})^{6}$ in powers of $x$ , is

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Solution

$\Rightarrow (1 + x + x^{2} + x^{3})^{6}$
$= {[(1 + x) + x^{2} (1 + x)]}^{6}$
$= (1 + x)^{6} (1 + x^{2})^{6}$
Coefficient of $x^{4}$
$=^{6} C_{0} \times^{6} C_{2} +^{6} C_{2} \times^{6} C_{1} +^{6} C_{4} \times^{6} C_{0}$
$= 1 \times 15 + 15 \times 6 + 15 \times 1 = 15 + 90 + 15 = 120$
$∴ Coefficient of x^{4} = 120$

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