Download 3.1 Permutations

Permutations
 Counting where order matters
 If you have two tasks T1 and T2 that are performed in
sequence. T1 can be performed in n ways. T2 can be
performed in n ways.
 The sequence T1 T2 can be performed in (n1)(n2 ) ways.
3 possible ways
to perform T1
4 possible ways to
perform T2
Possible ways of performing Task 1 then Task 2
3 4 = 12
This is called the multiplication principle of counting.
 Lets say you have a password that is one letter followed
by 3 digits. How many possible unique passwords can
you create?
 26 letters in the alphabet
 10 possible numbers (0-9) for each digit
letters x digit 1 x digit 2 x digit 3
26 x 10 x 10 x 10 = 26000
nPr symbolizes the number of permutations of n objects
taken r at a time.
If 1 ≤ r ≤ n then:
nPr is:
n x (n-1) x (n-2)…..(n-r+1)
1 at a time 2 at a time 3 at a time
The number of permutations of 4 objects taken 3 at a
time = 4 x 3 x 2 = 24
Taken at a time:
Taken 3 at time.
(n) x (n-1) x (n-2)
4 x 3 x 2 = 24
1
2
3
 The sequences 12, 43, 31, 24 and 21 are some
permutations of set A taken 2 at a time. This is noted
as 4P2
 The number of permutations of 4 objects taken 2 at a
time =
4 x 3
= 12
(n) x (n-1)
taken at a time 1
2
 To further define nPr, we have a formula
nPr = n!/(n-r)!
! Means factorial
3! = 1 x 2 x 3 = 6
 If we had a deck of cards, set A = 52 and 5 cards are
dealt. The number of permutations of A taken 5 at a
time is:
 52P5 = 52!/47! = 311,875,200
 Or
52 x 51 x 50 x 49 x 48
(n) x (n-1) x (n-2) x (n-3) x (n-4)
5
Taken at a time: 1
2
3
4
 How many words of three distinct letters can be
formed from the letters of the word MAST?
 4P3 = 4!/(4-3)! = 4!/1! = 24
 How many distinguishable words can be formed with
the letters of MISSISSIPPI?
 There are 11 letters
 1 M, 4 I, 4 S, 2 P
11!/(1!) (4!) (4!) (2!) = 39,916,800/(1)(24)(24)(2)
= 39,916,800/1152 = 34,650