Editing BinomialCoefficient
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Binomial coefficients can be defined in three equivalent ways. EQN:{n\choose{}r} is ... * ... the coefficient of EQN:x^r when you expand out EQN:(1+x)^n . ** This is where the name comes from. * ... the number of different ways to select /r/ objects from a set of /n./ ** This is why EQN:n\choose~r is sometimes pronounced "n choose r". *** See also: Combinations * ... defined by the following three facts: ** (1) EQN:n\choose~r is 0 whenever EQN:r<0 or EQN:r>n ** (2) EQN:n\choose~r is 1 whenever EQN:r=0 or EQN:r=n ** (3) EQN:{n+1\choose~r+1}={n\choose~r}+{n\choose~r+1} *** These are the equations that define Pascal's triangle. The interplay between these three quite different ways of thinking about the same objects leads to a great deal of beautiful mathematics. The binomial coefficients turn up (unsurprisingly) in the Binomial Theorem. ---- !! Enrichment Show that the "three facts" given above can be reduced to these two: * (1) EQN:0\choose~r is 1 if and only if /r=0/ and 0 otherwise * (2) EQN:{n+1\choose~r+1}={n\choose~r}+{n\choose~r+1}