Question

If the sum of the coefficients of all the positive even powers of $x$ in the binomial expansion of $\left(2 x^{3}+\frac{3}{x}\right)^{10}$ is $5^{10}-\beta \cdot 3^{9}$, then $\beta$ is equal to____

Answer 1

$T _{ r +1}={ }^{10} C _{ r }\left(2 x ^{3}\right)^{10- r }\left(\frac{3}{ x }\right)^{ r }$
$={ }^{10} C _{ r } 2^{10- r } 3^{ r } x ^{30-4 r }$
Put $r=0,1,2, \ldots .7$ and we get $\beta=83$