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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 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.

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!! 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}