Question

How many terms does the binomial expansion have ${(\frac{2}{3}x-\frac{3}{2x})}^6$

Answer 1

By binomial expansion

$(a+b{)}^n=\sum _{r=0}^n{}^n{C}_r{a}^{n-r}{b}^r$

Here, $a=\frac{2}{3}x,n=6\&b=-\frac{3}{2x}$

$\therefore {(\frac{2}{3}x-\frac{3}{2x})}^6=\sum _{r=0}^6{}^6{C}_r{\left(\frac{2}{3}x\right)}^{6-r}{(-\frac{3}{2x})}^r$

On simplifying, we get

${(\frac{2}{3}x-\frac{3}{2x})}^6=\frac{64x^6}{729}+\frac{729}{64x^6}-\frac{243}{8x^4}-\frac{32x^4}{27}+\frac{20x^2}{3}+\frac{135}{4x^2}-20$