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}