Permutation SS2 Further Mathematics Lesson Note

PERMUTATION. 

Permutation is defined as the number of arrangements of objects. The different orders of arrangement are important. 

E.g. Find the number of ways of arranging the letters pqr.

= Pqr, prq, qrp, qpr, rqp. The number of ways is 6 ways

Similarly, for 4 letters the number of arrangements is 24

In general, the number of different arrangements of n different objects is equal to n! (n factorial)

N! = n x (n-1) x (n-2) x … x 3×2 x 1×0! (But, 0! = 1)

Eg 1: Simplify the following:   

1. 5!
2. 7!

   3! 4!

Solution

1. 5! = 5x4x3x2x1 = 120
2. =    7x6x5x4!    = 7×5  = 35

   Find the number of ways of arranging the letters of the word MACHINE

Solution: 

There are seven different letters in the word MACHINE, therefore the number of permutations is 7!  = 7x6x5x4x3x2x1       = 5040 ways

ARRANGEMENT OF n-OBJECTS TAKEN r-OBJECTS

If we are interested in the number of ways 2 letters of a 4-lettered word can be arranged, then the npr is the permutation of n objects taking at a time

npr =  

Example: Evaluate:

(a)  8p3 (b) 11p9

Solution:

8P3 = 8!      = 8! = 8×7×6×5= 8×7×6

          (8-3)!    5!

= 336

b) 11P9= 11!

             (11-9)!      

= 11x10x9x8x7x6x5x4x3x2! =    19958400

2. In how many ways can three people be seated on eight seats in a row?

Solution:

1st seat can be occupied by any of the 8 = 8 ways

2nd seat can be occupied in 6 ways

Hence, the number of ways = 8x7x6 = 336 ways

Alternatively, n = 8, r = 3

nPr = n!   =  8!  = 8×7×6×5

       (n-r)!  (8-3)!         5!

= 8×7×6 = 336 ways