section 2.3: permutations when all objects are...
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Section 2.3 Permutations soln.notebook
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Section 2.3: Permutations when all Objects are Distinguishable
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Recall:
A Permutation is an arrangement of objects in a definite order, where each object appears only once in each arrangement.
Case A. Arranging n different objects: The number of permutations (arrangements) from a set of n different objects, where n of them are used in each arrangement, can be calculated using n!.
1. A sixdigit secret code number uses the digits 1 through 6 exactly once each. How many possibilities are there for the code number?
2. How many arrangements are possible using all of the letters in WHISTLER?
3. Visitors to a movie website will be asked to rank 28 different movies. The website will present the movies in a different order for each visitor to reduce bias in the poll. How many permutations of the movie list are possible?
Questions 1 – 5 on Worksheet 3: Permutations
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Case B. Repetitions allowed: Sometimes we are interested in arrangements that allow the use of items more than once. The number of permutations that can be created from a set of n objects using r objects in each arrangement, where repetition is allowed is nr.
1. There are 9 switches on a fuse box. Each switch has two possible positions, on or off. How many different arrangements are there?
2. How many three letters words can be created, if repetitions are allowed?
Questions 6 – 9 on Worksheet 3: Permutations
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Case C. Arranging a Subset of Items: Sometimes you will be given a bunch of objects and you want to arrange only a few of them.
Example: If a committee has eight members.
a. How many ways can the committee members be seated in a row?
In this case we want to find the number of ways of arranging all members of the committee(case A)......so answer is n!
b. How many ways can the committee select a president, vice president and secretary.?In this case, we only want to order a select few of the members of the committee. So......
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Another way of doing this is to use the nPr .
To find the number of permutations or arrangements that can be made from a set of n different objects where r of them are used in each arrangement use the following permutation formula:
nPr = where 0 # r # n,
· n is the total number of items · r is the number of items you want to order.
The above formula implies NO REPETITIONS.
When all available objects are used in each arrangement, n and r are equal, so the notation nPn is used and nPn = n!
nPn =
As a result, any algebraic expression that involves factorials is defined as long as the expression is greater than or equal to 0.
(n +4)! is only defined for n ≥ 4.
The above formula implies NO REPETITIONS.
Reminder!
Definition: 0! = 1
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Examples
1. A) How many 4 letter passwords can you make using the letters “Math”?
B) How many 3 letter passwords can you make using the letters “Math”?
C) How many 2 letter passwords can you make using the letters “Math”?
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4. Evaluate the following expressions.
2. How many three‐digit numbers can be made using the numbers 1 through 9 if no numbers can be repeated?
3. John is a contestant on a game show. He’ll win $10 000 if he touches the correct three coloured tiles on the board (shown below) in the correct order.
How many sequences are possible?
A) 12P3 B) 15P2 C) 8P8
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5. Written driver's test consists of 25 questions randomly selected from a pool of 100 questions. How many different versions of the test are there?
6. Three people chosen (1 + 2 alternates) from a group of 9 people.
a) How do you know it's a permutation?
Order is important. ONE person is chosen, they cannot be chosen again.
b) how many ways can you select the 3 people?
Homework: Page 93 – 95: Questions 1 – 8, 11.
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7. A password consists of 5digits from 0 to 9.
A) Is the order of the digits in a password important? Explain.
B) How many arrangements are possible if repetition is allowed?
C) How many arrangements are possible if repetition is not allowed? Do the number of choices stay the same or are they reduced?
D) In which case is there a greater number of permutations possible?
Homework:Page 93 – 95: Questions 1 – 8, 11.Questions 1013 on worksheet 3 permutations
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Case D. Specific Positions:
Frequently when arranging items, a particular position must be occupied by a particular item. The easiest way to approach these questions is by analyzing how many possible ways each space can be filled.
1. How many ways can you order the letters of KITCHEN if it must start with a consonant and end with a vowel?
2. A) How many numbers can be made from rearranging 2345678 if the number must begin with two odd digits?
B) How many numbers can be made from rearranging 2345678 if the number must begin with exactly two odd digits?
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Case E. Always Together:
Frequently, certain items must always be kept together. To do these questions ,
1) Treat the objects together as 1, and determine the number of arrangements
2) For each group that is together, find the number of “internal” arrangements
Examples:
1. How many arrangements of the word ACTIVE are there if C & E must always be together?
2. How many ways can 3 math books, 5 chemistry books, and 7 physics books be arranged on a shelf if the books of each subject must be kept together?
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3. How many arrangements of the word FAMILY exists if A and L must always be together?
4. Seven math students (Amy, Brady, Christopher, Dylan, Ellie, Frank, Gina) are going to stand in a line. How many ways can they stand if:
A) the girls must stay together?
B) Brady and Dylan must stay together?
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Case F. Never Together:
If certain items must be kept apart, you will need to figure out how many possible positions the separate items can occupy.
1. How many arrangements of the word active are there if C and E must never be together?
2. How many arrangements of daughter are there if none of the vowels can ever be together?
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3. Seven math students (Amy, Brady, Christopher, Dylan, Ellie, Frank, Gina) are going to stand in a line. How many ways can they stand if ...
A) Brady and Dylan can’t be together?
B) none of the boys can be together.
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Case G. More than one Case (Adding):
Given a set of items, it is possible to form multiple groups by ordering any 1 item from the set, any 2 items from the set, and so on. If you want the total arrangements from multiple groups you have to add the results of each case.
KEY WORDS: "at most", "at least" and "or".
1. How many 2, 3 and 4 letter words can be formed from the word CANS?
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2. To open the garage door of Mary's house, she uses a keypad containing the digits 0 through 9. The password must be at least a 4 digit code to a maximum of 6 digits, and each digit can only be used once in the code. How many different codes are possible?
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3. Tania needs to create a password for a social networking website she registered with. The password can use any digits from 0 to 9 and/or any letters of the alphabet. The password is case sensitive, so she can use both lower and uppercase letters. A password must be at least 5 characters to a maximum of 7 characters, and each character can be used only once in the password. How many different passwords are possible?
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Solving Equations of the form n Pr = k
• simplify the expression so that it no longer contains factorials
• solve the resulting equation
Ex. Solve.
A) nP2 = 30
C) n+ 1P2 = 20
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Ex. Mary has a set of posters to arrange on her bedroom wall. She can only fit two posters side by side. If there are 72 ways to choose and arrange two posters, how many posters does she have in total?
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A code consists of three letters chosen from A to Z and three digits chosen from 0 to 9, with no repetition of letters or numbers. How many codes are possible?
Final Example!!!!