4.1 Permutations Part 1


(A) rs Multiplication Principle/ Rule

1. If an operation can be carried out in r ways and another operation can be carried out in s ways, then the number of ways to carry out both the operations consecutively is r × s, i.e. rs.

2. The rs multiplication principle can be expanded to three or more operations. If the numbers of ways for the occurrence of events A, B and C are r, s and p respectively, the number of ways for the occurrence of all the three events consecutively is r × s × p, i.e. rsp.

Example 1:
There are 3 different roads to travel from town P to town Q and 4 different roads to travel from town Q to town R. Calculate the number of ways a person can travel from town P to town R via town Q.

Solution:
3 × 4 = 12


(B) Permutations


Example 2:
Calculate each of the following.
(a) 7!
(b) 4!6!
(c) 0!5!
( d ) 7 ! 5 ! ( e ) 8 ! 4 ! ( f ) n ! ( n 2 ) ! ( g ) n ! 0 ! ( n 1 ) ! ( h ) 3 ! ( n + 1 ) ! 2 ! n !


Solution:
(a) 7! = 7 × 6 × 5 × 4 × 3 × 2 × 1 = 5040
(b) 4!6! = (4 × 3 × 2 × 1)( 6 × 5 × 4 × 3 × 2 × 1) = 17280
(c) 0!5! = (1)( 5 × 4 × 3 × 2 × 1) = 120

(d) 7 ! 5 ! = 7 × 6 × 5 ! 5 ! = 7 × 6 = 42 (e) 8 ! 4 ! = 8 × 7 × 6 × 5 × 4 ! 4 ! = 8 × 7 × 6 × 5 = 1680 (f) n ! ( n 2 ) ! = n ( n 1 ) ( n 2 ) ( n 2 ) = n ( n 1 ) (g) n ! 0 ! ( n 1 ) ! = n ( n 1 ) ( 1 ) ( n 1 ) = n (h) 3 ! ( n + 1 ) ! 2 ! n ! = 3 × 2 ! ( n + 1 ) ( n ) ( n 1 ) 2 ! n ( n 1 ) = 3 ( n + 1 )

Calculator Computation: 



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