$\Rightarrow np=20 \dots \left(\mathrm{i}\right)$

and standard deviation of a binomial distribution $=4$

$\Rightarrow np\left(1-p\right)=4 \dots \left(\mathrm{ii}\right)$

Put $np=20$ from equation $\left(\mathrm{i}\right)$ to equation $\left(\mathrm{ii}\right)$, we get

$\Rightarrow 20\left(1-p\right)=4$

$\Rightarrow p=\frac{4}{5}$

Put the value of $p$ in the equation $\left(\mathrm{i}\right)$, we get

$\Rightarrow n\left(\frac{4}{5}\right)=20$

$\Rightarrow n=25$