Question

If the coefficient of $x^{10}$ in the binomial expansion of ${\left(\frac{\sqrt{x}}{5^{\frac{1}{4}}}+\frac{\sqrt{5}}{x^{\frac{1}{3}}}\right)}^{60}$ is $5^{k}l$, where $l,k\in N$ and $l$ is coprime to $5$, then $k$ is equal to_____.

Answer 1

Answer: 5

${\left(\frac{\sqrt{x}}{5^{1/4}}+\frac{\sqrt{5}}{x^{1/3}}\right)}^{60}$
$T_{r+1}={}^{60}{C}_r{\left(\frac{x^{1/2}}{5^{1/4}}\right)}^{60-r}\left(\frac{5^{1/2}}{x^{1/3}}\right)r$
$={}^{60}{C}_{r}5\frac{3r-60}{4}.x\frac{180-5r}{6}$
$\frac{180-5r}{6}=10\Rightarrow r=24$

Powers of 5 in $={}^{60}{C}_{24}\cdot 5^3=\frac{5^{14}}{5^4\times 5^8}\times 5^3=5^5$