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

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

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

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

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

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

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

vii) If $n$ is an even natural number, then the greatest among $^{n} \rm C_{0}, ^{n} \rm C_{1}, ^{n} \rm C_{2}, \ldots , ^{n} \rm C_{n}$ is  $^{n} \rm C_{\frac{n}{2}}$

If $n$ is odd natural number, then the greatest among $^{n} \rm C_{0}, ^{n} \rm C_{1}, ^{n} \rm C_{2}, \ldots , ^{n} \rm C_{n}$ is  $^{n} \rm C_{\frac{n-1}{2}}$ 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 $\frac{n!}{(n-r)!r!}$ $= ^{n} \rm C_{r}$