Question:
The middle term in the expansion of $\left(\frac{2 x^{2}}{3}+\frac{3}{2 x^{2}}\right)^{10}$ is
(a) 251
(b) 252
(c) 250
(d) none of these
Solution:
(b) 252
Here, $n$, i. e., 10 , is an even number.
$\therefore$ Middle term $=\left(\frac{10}{2}+1\right)$ th term $=6$ th term
Thus, we have
$T_{6}=T_{5+1}$
$={ }^{10} C_{5}\left(\frac{2 x^{2}}{3}\right)^{10-5}\left(\frac{3}{2 x^{2}}\right)^{5}$
$=\frac{10 \times 9 \times 8 \times 7 \times 6}{5 \times 4 \times 3 \times 2} \times \frac{2^{5}}{3^{5}} \times \frac{3^{5}}{2^{5}}$
$=252$