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  4. In the expansion of (1+x+x3+x4)10, the coefficient of x4 is

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Q. In the expansion of $\left(1+x+x^{3}+x^{4}\right)^{10}$, the coefficient of $x^{4}$ is

Binomial Theorem

Solution:

$\left(1+x+x^{3}+x^{4}\right)^{10}=\left[(1+x)\left(1+x^{3}\right)\right]^{10}$
$=(1+x)^{10}\left(1+x^{3}\right)^{10} $
$=\left(1+{ }^{10} C_{1} x+{ }^{10} C_{2} x^{2}+{ }^{10} C_{3} x^{3}+{ }^{10} C_{4} x^{4} \ldots\right)$
$ \times\left(1+{ }^{10} C_{1} x^{3}+{ }^{10} C_{2} x^{6} \ldots\right) $
$\therefore $ Coefficient of $x^{4}=\left({ }^{10} C_{1}\right)\left({ }^{10} C_{1}\right)+{ }^{10} C_{4} $
$= 100+\frac{10 \cdot 9 \cdot 8 \cdot 7}{1 \cdot 2 \cdot 3 \cdot 4}$
$=100+210=310 $