Probability & Statistics Section 3.4

 The letters a, b, can c can be arranged in six different orders: abcbaccab acbbcacba  Each of these arrangements is called a permutation of the letters a, b, and c.  An arrangement of a set of objects in a definite order is called a permutation of the objects

 Any one of the three letters, a, b, or c, may be written first, as indicated by:  After the first letter has been chosen, the second must be selected from the remaining 2 letters:  Only one selection remains for the last letter:

 Using the fundamental counting principle, the number of permutations is found: 3 x 2 x 1 = 6 permutations  3 x 2 x 1 = 3! (read as “three factorial) 321

Example : How many different arrangements can be formed from the letters in the word JUSTICE using all 7 letters? A permutation is an ordered arrangement of objects. The number of different permutations of n distinct objects is n!. “ n factorial” n ! = n · ( n – 1)· ( n – 2)· ( n – 3)· …· 3· 2· 1 7! = 7 · 6 · 5 · 4 · 3 · 2 · 1 = 5040 arrangements

 How many permutations can be formed from the letters in the word JUSTICE using only 5 letters at a time? ____ ____ ____ ____ ____

The number of permutations of n elements taken r at a time is # in the group # taken from the group Example: You are required to read 5 books from a list of 8. In how many different orders can you do so?

8!

9! 7!

5! (5 – 3)!

100! 98!

4P14P1

6P26P2

3P33P3

10 P 3

100 P 2

5! 0!

4! (4 – 0)!

5P05P0

How many 4 letter permutations can be formed from the word iphone? 6 P 4 = 360

 How many permutations of the letters a, a, and b are there? aabababaa  However, since you have repeated arrangements, there are really only three distinguishable (unique) arrangements. Like elements

 How many permutations can be formed from the letters, taken 5 at a time, in DADDY? 20

 How many permutations can be formed from the letters, taken 4 at a time, in NOON? 6

How many ways can {1, 2, 3} and {1, 2, 3, 4} be arranged in a circle? P n = (n – 1)!

 How many ways can 12 horses be arranged on a merry-go-round? 39,916,800

 How many different arrangements of the word PENCIL are there? 6! = 720

 How many arrangements of PENCIL have N as the first letter? 5! = 120

 How many arrangements of PENCIL have N as the last letter? 5! = 120

 How many arrangements of PENCIL will have PEN together? 4! 3! = 144

 How many arrangements of ITALY have ITA together? 3! 3! = 36