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Will Orrick (talk | contribs) →Example: Extend example to include the multisets of the proof of Theorem two. |
Will Orrick (talk | contribs) →Theorem two: missing "the"; "cardinality" --> "size" |
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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
:<math>\left(\!\!{n+1\choose k-1}\!\!\right) = \left(\!\!{k\choose n}\!\!\right) = \binom{n + k - 1}{k - 1} = \binom{10+4-1}{4 - 1} = \binom{13}{3} = 286,</math>
where the [[Multiset#Counting multisets|multiset coefficient]] <math>\left(\!\!{k\choose n}\!\!\right)</math> is the number of multisets of
This corresponds to [[Composition (combinatorics)|weak compositions]] of an integer.
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