Question

Find the coefficient of $\frac{1}{y^{10}}$ in the expansion of ${(y^3+\frac{a^7}{y^5})}^{10}$

Answer 1

Let the $(r+1{)}^{th}$ term contain the term $\frac{1}{y^{10}}$.

$T_{r+1}{=}^{10}{C}_r(y^3{)}^r{\left(\frac{a^7}{y^5}\right)}^{10-r}$

${=}^{10}{C}_r{a}^{7(10-r)}(y{)}^{3r-5(10-r)}\frac{1}{y^{10}}=y^{-10}$

$3r-5(10-r)=-10$

$3r-50+5r=-10$

$8r=40=r=5$

Coefficient ${=}^{10}{C}_5{a}^{35}$.