The middle term in the binomial expansion ${(\frac{2x^2}{3}+\frac{3}{2x^2})}^{10}$  is  

in the binomial expansion ${(\frac{2x^2}{3}+\frac{3}{2x^2})}^{10}$  is  $\left(\frac{10}{2}+1\right)\mathrm{th} \mathrm{term}$

$\begin{array}{r}\begin{array}{rl}{T}_6& {=}^{10}{C}_5{\left(\frac{2x^2}{3}\right)}^{10-5}{\left(\frac{3}{2x^2}\right)}^5\\ & \left(\because T_{r+1}{=}^n{C}_r(a{)}^{n-r}{b}^r\right).\\ & =\frac{10!}{5!5!}\frac{2^5{x}^{10}}{3^5}\cdot \frac{3^5}{2^5\cdot x^{10}}=\frac{10!}{5!5!}=\frac{10\times 9\times 8\times 7\times 6}{5\times 4\times 3\times 2\times 1}\end{array}\\ =252\text{. }\end{array}$