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 to base 10. We give a generalization of Champernowne's construction to various other digit systems, including generalized Lüroth 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.