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.

## Enrichment

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) ${n+1\choose~r+1}={n\choose~r}+{n\choose~r+1}$

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