6.3 Combinations

6.3 Combinations

(1) The number of combinations of r objects chosen from n different
      objects is given by :
n C r = n! r!(nr)!

(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.
Note: (i)   n C 0 =1 (ii)   n C n =1 (iii)   n C r = n C nr 

Example 1:
Calculate the value of  7 C 2 7 C 2 = 7! ( 72 )! ×2! = 7! 5! ×2! = 7 ×6 ×5! 5! ×2! = 7×6 2×1 =21
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  =   6 C 3 ×   3 C 3 =20×1 =20