If the sum of the co-efficients of all the positive even powers of...

Given,

Binomial expression ${\left(2{x}^{3} + \dfrac{3}{x}\right)}^{10}$, so its ${n}^{th}$ term is given by,

${T}_{r + 1} = Cr 10 {\left(2{x}^{3}\right)}^{10 - r}{\left(\dfrac{3}{x}\right)}^{r}$

$= Cr 10 {2}^{10 - r}{3}^{r}{x}^{30 - 4r}$

Now putting $r = 0, 1, 2, . . . . . 7$ and adding all term we get

${\left(2{x}^{3} + \dfrac{3}{x}\right)}^{10} = C0 10 {2}^{10}{3}^{0}{x}^{30} + C1 10 {2}^{9}{3}^{1}{x}^{26} . . . . . . . . . . . . . . C7 10 {2}^{3}{3}^{7}{x}^{2} + C8 10 {2}^{2}{3}^{8}\dfrac{1}{{x}^{2}} . . . .$

${\left(2{x}^{3} + \dfrac{3}{x}\right)}^{10} - C8 10 {2}^{2}{3}^{8}\dfrac{1}{{x}^{2}} . . . . . . . = C0 10 {2}^{10}{3}^{0}{x}^{30} + C1 10 {2}^{9}{3}^{1}{x}^{26} . . . . . . . . . . . . . . C7 10 {2}^{3}{3}^{7}{x}^{2}$

Put $x = 1$ both side we get,

${5}^{10} - {3}^{9}\left(60 + 20 + 3\right) = C0 10 {2}^{10}{3}^{0} + C1 10 {2}^{9}{3}^{1} . . . . . . . . . . . . . . C7 10 {2}^{3}{3}^{7}$

${5}^{10} - {3}^{9}\left(83\right) = \text{sum of even coefficients}$

So $\beta = 83$