Permutations
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A permutation  is an arrangement of objects in specific order. 
The order of the arrangement is
important!! 

Consider, four students walking toward their school entrance.  How many different ways could they arrange themselves in this side-by-side pattern?

1,2,3,4       2,1,3,4       3,2,1,4       4,2,3,1
1,2,4,3       2,1,4,3       3,2,4,1       4,2,1,3
1,3,2,4       2,3,1,4       3,1,2,4       4,3,2,1
1,3,4,2       2,3,4,1       3,1,4,2       4,3,1,2
1,4,2,3       2,4,1,3       3,4,2,1       4,1,2,3
1,4,3,2       2,4,3,1       3,4,1,2       4,1,3,2

The number of different arrangements is 24 or 4! = 4 3 2 1.    There are 24 different arrangements, or permutations, of the four students walking side-by-side.
 

The notation for a permutation: 
n Pr
  
n  is the total number of objects 
  
r   is the number of objects chosen (want)

 

See how to use your
TI-83+/TI-84+ graphing calculator  with permutations.
Click calculator.

The formula for a permutation:
           

 

OR

The formula for a permutation:
                          
                    (Remember that 0! = 1.)
(Note:   if  n = r, as it did with the students walking side-by-side, then   n Pr n!  for either formula.)


Examples: 

1.    Compute:  5 P 5         5 4 3 2 1  =  120
2.    Compute:   6 P 2       6 5  =  30                or    
                                     multiply by two factors
                                   of the factorial, starting with 6
3.    Find the number of ways to arrange 5 objects that are chosen from a set of 7 different objects.
        7
P 5 =   7
6543  =  2520      or     
      
4.  What is the total number of possible 5-letter arrangements of the letters  w, h, i, t, e,  if each letter is used only once in each arrangement? 
         
5 P5   =   54321   =   120     or           or    simply  5!
        
5.   How many different 3-digit numerals can be made from the digits  4, 5, 6, 7, 8   if a digit can appear just once in a numeral?
       
5 P3  =   543  =  60            or       
     
  


Permutations with Special Arrangements:

Example:  Using the letters in the word  " square ", tell how many 6-letter arrangements, with no repetitions, are possible if the :
           a)  first letter is a vowel.
           b)  vowels and consonants alternate, beginning with a consonant.


Solution:

Part a:  Hint:  When working with "arrangements", it is often helpful to put lines down to represent the locations of the items. 
       For this problem, six "locations" are needed for 6-letter arrangements.
                    
     _____     _____    _____    _____    _____    _____

The first locations must be a vowel (u, a, e).  There are three ways to fill the first location.
                             __3___     _____   _____   _____   _____   _____

After the vowel has been placed in the first location, there are 5 letters left to be arranged in the remaining five spaces.

                          __3__ __5__  __4__  __3__    __2__ __1__      or
                        
                        
    3   5P5    =     3 120  =  360

Part b:   Six locations are needed for the 6-letter arrangements.
                         
_____     _____   _____   _____   _____   _____
 Beginning with a consonant, every other location must be filled with a consonant. (s, q, r )    
                       __3__   _____  __2___  _____   ___1__  _____

The remaining locations are filled with the remaining three vowels:
              
   __3__   __3___  __2___  __2___   ___1__ ___1__  =  36





 
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