Binomial Theorem

The constant term of $\left(x+x^{-\frac{3}{2}}\right)^{15}$ when expanded is $5005$.

When the above expression is expanded, we get 

Applying Binomial Therom:

$\sum _{i=0}^{15}\binom{15}{i}x^{\left(15-i\right)}\left(x^{-\frac{3}{2}}\right)^i$

$x^{15}+15x^{\frac{25}{2}}+105x^{10}+455x^{\frac{15}{2}}+1365x^5+3003x^{\frac{5}{2}}+5005+\frac{6435}{x^{\frac{5}{2}}}+\frac{6435}{x^5}+\frac{5005}{x^{\frac{15}{2}}}+\frac{3003}{x^{10}}+\frac{1365}{x^{\frac{25}{2}}}+\frac{455}{x^{15}}+\frac{105}{x^{\frac{35}{2}}}+\frac{15}{x^{20}}+\frac{1}{x^{\frac{45}{2}}}$

The constant term as seen above is $5005$.

-Vinculum