  Champernowne famously proved that the number
$0.(1)(2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)...$ formed by concatenating all the
integers one after another is normal base 10. We give a generalization of
Champernowne's construction to various other digit systems, including
generalized L\"uroth series with a finite number of digits. For these systems,
our construction simplifies a recent construction given by Madritsch and Mance.
Along the way we give an estimation of the sum of multinomial coefficients
above a tilted hyperplane in Pascal's simplex, which may be of general
interest.
