Transcript Section 11.2 - vincentsmathpage

Section 11.2
Permutations
Definitions
• A permutation from a group of items occurs when no
item is used more than once and the order of
arrangement makes a difference.
• Factorial Notation
n! = n(n-1)(n-2)…(3)(2)(1)
0! = 1
• Premutations Formula - the number of permutations
possible if r items are taken from n items is:
n!
n Pr 
(n  r )!
• Permutations of Duplicate Items – The numer of
permutations of n items, where p items are identical, q
items are identical, r items are identical, and so on, is:
n!
p!q!r !...
Example 1
• Evaluate:
7! 7  6  5  4  3  2 1  5040
16! 16 15 14 13  ... 1 16 15

 240

14 13  ... 1
1
14!
P 
11 3
11 10  9  8  7  ... 1
11!


8  7  ... 1
(11  3)!
11 10  9
 990
1
Example 2
• In how many ways can Billy, Quan, Millz,
Taryn, and Julietta line up?
We have five people and we
want to line all five of them up, so
we have 5P5.
5!
Remember
P

5 5
0! = 1
(5  5)!
5  4  3  2 1

 120
1
Example 3
• You have T-Mobile and you are trying to decide who to put
into your Fave Five. You decided you would rank your
friends to help you decide. If you have 20 numbers in your
phone, how many ways can the five favorites be ranked?
We have 20 people and we want to rank
the top 5, so we need to find 20P5.
20!
20 P5 
(20  5)!
20 P5  1,860, 480
You can do
this on your
calculator
too!
Example 4
• Use the formula for the number of permutations of duplicate
items to solve:
– In how many distinct ways can the letters of the word
BANANA be arranged?
There are six letters in the word, so n = 6, but A repeats
three times, so p = 3, and N repeats twice, so q = 2.
n!
6! 6  5  4  3  2 1


p !q ! 3!2! (3  2 1)(2 1)
720
 60
12
Example 5
• Brad asked a girl for her number, but she didn’t want to give it
to him. She did however tell him that her phone number has
two 6’s, three 4’s, and two 3’s. How many phone numbers
would Brad have to try if he calls every possible phone
number?
Since phone numbers have 7 digits, n = 7, but 6 repeats
twice, so p = 2, 4 repeats three times, so q = 3, and 3
repeats twice, so r = 2.
n!
7!
7 * 6 *5* 4 *3* 2 *1


p !q !r ! 2!3!2! (2 *1)(3* 2 *1)(2 *1)
 210
Homework
• Pages 343 – 346 evens only.