6.1 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! ( n2 )! (g) n!0! ( n1 )! (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! ( n2 )! = n( n1 )( n2 ) ( n2 ) =n( n1 ) (g)  n!0! ( n1 )! = n( n1 )( 1 ) ( n1 ) =n (h)  3!( n+1 )! 2!n! = 3×2!( n+1 )( n )( n1 ) 2!n( n1 ) =3( n+1 )


Calculator Computation: