In this case, the representation of a tuple as a sequence of stars and bars, with the bars dividing the stars into bins, is unchanged. The weakened restriction of nonnegativity (instead of positivity) means that one may place multiple bars between two stars, as well as placing bars before the first star or after the last star. Thus, for example, when n = 7 and k = 5, the tuple (4, 0, 1, 2, 0) may be represented by the following diagram.
Fig. 3: four bars give rise to five bins containing 4, 0, 1, 2, and 0 objects
This establishes a one-to-one correspondence between tuples of the desired form and selections with the replacement of k − 1 gaps from the n + 1 available gaps, or equivalently (k − 1)-element multisets drawn from a set of size n + 1. By definition, such objects are counted by the multi choose number.
To see that these objects are also counted by the binomial coefficient, observe that the desired arrangements consist of n + k − 1objects (n stars and k − 1 bars). Choosing the positions for the stars leaves exactly k − 1 spots left for the k − 1 bars. That is, choosing the positions for the stars determines the entire arrangement, so the arrangement is determined with {\displaystyle selections. Note that , reflecting the fact that one could also have determined the arrangement by choosing the positions for the k − 1 bars.