Properties of Combinations

Standard Properties

1. Symmetry: $^{n}C_{r} = {}^{n}C_{\,n-r}$. Choosing which $r$ to include is the same as choosing which $(n-r)$ to leave out.
2. End values: $^{n}C_{0} = {}^{n}C_{n} = 1$ (one way to choose none, one way to choose all), and $^{n}C_{1} = {}^{n}C_{\,n-1} = n$.
3. Pascal's rule: $^{n}C_{r} + {}^{n}C_{r-1} = {}^{n+1}C_{r}$.
4. Equality condition: if $^{n}C_{x} = {}^{n}C_{y}$ then either $x = y$ or $x + y = n$.

Total Number of Selections (any size)

The number of ways to select at least one object from $n$ distinct objects, of any size from $1$ to $n$, is

$^{n}C_{1} + {}^{n}C_{2} + \cdots + {}^{n}C_{n} = 2^{n} - 1.$ …