BinomialcoecientsDe nitionForn12andk01nnkn knkNotethatbyde nition01AlternatenotationsnCkorCnkAlternatede nitionnknn1nk1 kThisversionisconvenien ID: 206965
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Math408,ActuarialStatisticsIA.J.Hildebrand BinomialcoecientsDenition:Forn=1;2;:::andk=0;1;:::;n,nk=n! k!(n�k)!.(Notethat,bydenition,0!=1.)Alternatenotations:nCkorC(n;k)Alternatedenition:nk=n(n�1):::(n�k+1) k!.(Thisversionisconvenientforhand-calculatingbinomialcoecients.)Symmetryproperty:nk=nn�kSpecialcases:n0=nn=1;n1=nn�1=nBinomialTheorem:(x+y)n=nXk=0nkxkyn�kBinomialTheorem,specialcase:nXk=0nkpk(1�p)n�k=1CombinatorialInterpretations:nkrepresents1.thenumberofwaystoselectkobjectsoutofngivenobjects(inthesenseofunorderedsampleswithoutreplacement);2.thenumberofk-elementsubsetsofann-elementset;3.thenumberofn-letterHTsequenceswithexactlykH'sandn�kT's.Binomialdistribution:Givenapositiveintegernandanumberpwith0p1,thebinomialdistributionb(n;p)isthedistributionwithdensity(p.m.f.)f(k)=nkpk(1�p)n�k,fork=0;1;:::;n.2