jmc

We have $$\begin{array}{rl}{2}^{2001}\times 5^{1950}÷4^{27}& =2^{2001}÷2^{54}\times 5^{1950}\\ & =2^{1947}\times 5^{1950}\\ & =(2\times 5{)}^{1947}\times 5^3\\ & =125\times 10^{1947}\end{array}$$Since $125\times 10^{1947}$ has three non-zero digits followed by 1947 zeros, it has a total of $\boxed{1950}$ digits.