Most recent change of BinomialCoefficient

Edit made on May 29, 2016 by ColinWright at 20:30:28

Deleted text in red / Inserted text in green

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
** This
is where the name comes from.) from.
* ... the number of different ways to select /r/ objects from a set of /n./ (This
** This
is why EQN:n\choose~r is sometimes pronounced "n choose r".) (See combinations) 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 These are the equations that define Pascal's triangle.) 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}