Binomial coefficients can be defined in three equivalent ways. ${n\choose{}r}$ is ...
- ... the coefficient of $x^r$ when you expand out $(1+x)^n$ . (This is where the name comes from.)
- ... the number of different ways to select r objects from a set of n. (This is why $n\choose~r$ is sometimes pronounced "n choose r".) (See combinations)
- ... defined by the following three facts:
- (1) $n\choose~r$ is 0 whenever $r<0$ or $r>n$
- (2) $n\choose~r$ is 1 whenever $r=0$ or $r=n$
- (3) ${n+1\choose~r+1}={n\choose~r}+{n\choose~r+1}$
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.
Show that the "three facts" given above can be reduced to these two:
- (1) $0\choose~r$ is 1 if and only if r=0 and 0 otherwise
- (2) ${n+1\choose~r+1}={n\choose~r}+{n\choose~r+1}$