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}

Discover more from ICSE / ISC / CBSE Mathematics Portal for K12 Students

Subscribe to get the latest posts sent to your email.