Stars and bars (combinatorics): Difference between revisions

Revision as of 08:05, 30 January 2025

Revision as of 07:03, 22 February 2025

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If, for example, there are two balls and three bins, then the number of ways of placing the balls is <math>\tbinom{2+3-1}{3-1} = \tbinom{4}{2} = 6</math>. The table shows the six possible ways of distributing the two balls, the strings of stars and bars that represent them (with stars indicating balls and bars separating bins from one another), and the subsets that correspond to the strings. As two bars are needed to separate three bins and there are two balls, each string contains two bars and two stars. Each subset indicates which of the four symbols in the corresponding string is a bar.

{| class="wikitable"

|+ Six configurations of two balls in three bins and their star and bar represetationsrepresentations

|-

! Bin 1 !! Bin 2 !! Bin 3 !! String !! Subset of {1,2,3,4}