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Will Orrick (talk | contribs) Reorder statements in Theorem two to match what's in the proof. Typo corrections. Elaborate on different multiset interpretations and how this corresponds to interchanging bars and stars. |
Will Orrick (talk | contribs) →Theorem two: Move links to earlier occurrences. Theorem two now states the binomial coefficient formula, for those not familiar with multisets and multichoose. |
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===Theorem two===
For any pair of positive integers {{mvar|n}} and {{mvar|k}}, the number of {{mvar|k}}-[[tuple]]s of '''non-negative''' integers whose sum is {{mvar|n}} is equal to the number of
:<math>\binom{n + k - 1}{k - 1}.</math>
For example, if {{math|1=''n'' = 10}} and {{math|1=''k'' = 4}}, the theorem gives the number of solutions to {{math|1=''x''{{sub|1}} + ''x''{{sub|2}} + ''x''{{sub|3}} + ''x''{{sub|4}} = 10}} (with {{math|''x''{{sub|1}}, ''x''{{sub|2}}, ''x''{{sub|3}}, ''x''{{sub|4}} <math>\ge0</math> }}) as:
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