  In a multi-base representation of an integer (in contrast to, for example,
the binary or decimal representation) the base (or radix) is replaced by
products of powers of single bases. The resulting numeral system has desirable
properties for fast arithmetic. It is usually redundant, which means that each
integer can have multiple different digit expansions, so the natural question
for the number of representations arises. In this paper, we provide a general
asymptotic formula for the number of such multi-base representations of a
positive integer $n$. Moreover, we prove central limit theorems for the sum of
digits, the Hamming weight (number of non-zero digits, which is a measure of
efficiency) and the occurrences of a fixed digits in a random representation.
