# 4.3 Combinations

4.3 Combinations
(1) The number of combinations of r objects chosen from n different objects is given by :
$\overline{){}^{n}{C}_{r}=\frac{n!}{r!\left(n-r\right)!}}$
(2) A combination of r objects chosen from n different objects is a selection of a set of r objects chosen from n objects. The order of the objects in the chosen set is not taken into consideration.
$\overline{)\begin{array}{l}Note:\\ \left(i\right){\text{}}^{n}{C}_{0}=1\\ \left(ii\right){\text{}}^{n}{C}_{n}=1\\ \left(iii\right){\text{}}^{n}{C}_{r}{=}^{n}{C}_{n-r}\end{array}}$

Example 1:

Calculator Computation:

Example 2:

There are 6 marbles, each with different colour, which are to be divided equally between 2 children. Find the number of different ways the division of the marbles can be done.

Solution:
Number of ways giving 3 marbles to the first child =  ${}^{6}{C}_{3}$

Number of ways giving the remaining 3 marbles =  ${}^{3}{C}_{3}$

So, the number of different ways the division of the marbles
$\begin{array}{l}={\text{}}^{6}{C}_{3}×{\text{}}^{3}{C}_{3}\\ =20×1\\ =20\end{array}$