A combination is a selection of all or part of a set of objects, without regard to the order in which objects are selected.

If $n$ is a natural number and $r$ is a non-negative integer such that $0 \leq r \leq n$, then

$\displaystyle \text{i) } ^{n} \rm C_{r} = \frac{n!}{(n-r)! r!}$

$\displaystyle \text{ii) } ^{n} \rm C_r \times r! = ^{n} \rm P_r$

$\displaystyle \text{iii) } ^{n} \rm C_r = ^{n} \rm C_{n-r}$

$\displaystyle \text{iv) } ^{n} \rm C_r + ^{n} \rm C_{r-1} = ^{n+1} \rm C_r$

$\displaystyle \text{v) } ^{n} \rm C_r = \frac{n}{r} ^{n-1} \rm C_{r-1} = \frac{n}{r} \times \frac{n-1}{r-1} \cdot ^{n-2} \rm C_{r-2}$

$\displaystyle \text{vi) } ^{n} \rm C_{x} = ^{n} \rm C_{y} \Rightarrow x = y \text{ or } x+y = n$

vii) If $n$ is an even natural number, then the greatest among

$\displaystyle ^{n} \rm C_{0}, ^{n} \rm C_{1}, ^{n} \rm C_{2}, \ldots , ^{n} \rm C_{n} \text{ is } ^{n} \rm C_{\frac{n}{2}}$

If $n$ is odd natural number, then the greatest among

$\displaystyle ^{n} \rm C_{0}, ^{n} \rm C_{1}, ^{n} \rm C_{2}, \ldots , ^{n} \rm C_{n} \text{ is } ^{n} \rm C_{\frac{n-1}{2}} \text{ or } ^{n} \rm C_{\frac{n+1}{2}}$

The number of ways of selecting $r$ items or objects from a group of $n$ distinct objects is  $\displaystyle \frac{n!}{(n-r)!r!} = ^{n} \rm C_{r}$