If $\left(1+x+x^ {2}+x^ {3}\right)^ {5}=a_{0}+a_{1}x+a_{2}x^ {2}+....+a_{15}x^ {15}$, then $a_{10}$ equals

$\Rightarrow 1+x+x^{2}+x^{3}=\left[1(1+x)+x^{2}(1+x)\right]$

then $\left(1+x+x^{2}+x^{3}\right)^{5}=\left[(1+x)\left(1+x^{2}\right)\right]^{5}=(1+x)^{5}\left(1+x^{2}\right)^{5}$

$=(1+x)^{5} \times\left[{ }^{5} C_{0} 1^{5}+{ }^{5} C_{1} 1^{4} x^{2}+{ }^{5} C_{2} 1^{3} x^{4}+{ }^{5} C_{3} 1^{2} x^{6}+{ }^{5} C_{4} 1^{1} x^{8}+{ }^{5} C_{5} 1^{0} x^{10}\right]$

$a_{10}=\text { Coefficient of } x^{10}=\left({ }^{5} C_{0} 1^{5} \times{ }^{5} C_{5} 1^{0}\right)+\left({ }^{5} C_{2} 1^{3} \times{ }^{5} C_{4} 1^{1}\right)+\left({ }^{5} C_{4} 1^{1} \times{ }^{5} C_{3} 1^{2}\right)$

$=1+50+50$

So, $a_{10}=101$