4.3 Combinations

4.3 Combinations
(1) The number of combinations of r objects chosen from n different objects is given by :
n C r = n ! r ! ( n r ) !
(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.
N o t e : ( i ) n C 0 = 1 ( i i ) n C n = 1 ( i i i ) n C r = n C n r

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

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