LAST MATH CHAPTER EVER - Weeblymsbourgeois2016.weebly.com/uploads/5/5/9/1/...Section 9.1...

Section 9.1 Permutationssoln.notebook

1

May 29, 2017

Unit 9 Permutations, Combinations and Binomial Theorem

LAST MATH CHAPTER EVER

....UNTIL NEXT YEAR


2

May 29, 2017

Section 9.1: Permutations

Goals:Ø Using Graphic Organizers to count total possibilitiesØ Develop the Fundamental Counting Principal and apply it to solve problemsØ Solve simple counting problems with and without conditionsØ Use Graphic Organizers to count ordered arrangementsØ Determine the number of permutations of n different elements taken n at a timeØ Determine the number of permutations of n different elements taken r at a timeØ Simplify factorial expressions & determine value of 0!Ø Solve problems with (i) nPr and (ii) nPr & FCPØ Solve permutation equations of the form nPr = k for n or rØ Solve permutations with two or more identical elements (for n elements taken

n at a time)Ø Solve permutations with constraints when:

o (i) two or more objects are placed togethero (ii) two or more objects are NOT placed togethero (iii) certain objects in certain positions


3

May 29, 2017

Counting methods Methods used to determine the number of items in a specific set or number of ways something can be arranged

Includes:> Graphic Organizers> Fundamental Counting Principle> Factorial > Permutation> Combination


4

May 29, 2017

1. The school cafeteria offers a lunch combo for $6 where a person can order 1 sandwich (chicken, turkey, or grilled cheese), 1 side (fruit, yogurt, or soup), and 1 drink (juice or milk). Complete the tree diagram to determine the number of possible lunch combos.

(i) Using Graphic Organizers to Count Total # of Possibilities

Counting methods can be used visually to display the number of outcomes from two or more events. (Using tree diagrams, tables or lists)


5

May 29, 2017

(II) Using the Fundamental Counting Principal to Solve Problems

The Fundamental Counting Principle is the means to find the number of ways of performing two or more operations together.

The Fundamental Counting Principle:

• If we can perform a first task in x different ways AND

• If we can perform a second task in y different ways AND

• If we can perform a third task in z different ways, AND so on . . .

Then the first task followed by the second and so on can be performed in xyz⋅⋅. . . different ways.

** Note: FCP can't be used with OR!

Fundamental Counting Principle

= (# possibilities in event 1)(# possibilities in event 2)(# possibilities in event 3)...


6

May 29, 2017

2. A toy manufacturer makes a wooden toy in three parts. Determine how many different coloured toys can be produced?

Part 1: The top part may be coloured red or bluePart 2: The middle part may be orange, white, or blackPart 3: The bottom part may be yellow or green


7

May 29, 2017

(III) Solve Simple Counting Problems with & without conditions

Counting without restrictions

1. A store manager has selected 4 possible applicants for two different positions at a department store. In how many ways can the manager fill the positions?

# choices for position 1 and # of choices for position 2

2. How many license plates can be made using 3 letters (which can repeat) and 3 digits (which can repeat)?

3. How many ways can the letters in the word PENCIL be arranged?Idea: We have 6 objects and 6 possible positions to occupy

4. How many 3 digit numbers are there?


8

May 29, 2017

Counting with restrictions

Deal with restrictions first. Apply the FCP

5. How many 3 digit numbers are possible if repeats are not permitted?

6. In how many ways can five black cars and four red cars be parked next to each other in a parking garage if a black car has to be first and a red car has to be last?

Strategy


9

May 29, 2017

7. In NL a license plate consists of 3 letters followed by 3 numbers.A) How many license plate arrangements are possible if no letter or digit can be repeated?

B) How many license plates are possible if vowels (a, e, i, o, u) are not allowed?

8. How many ways can 4 girls and 3 boys sit in a row of 7 seats if a boy must be positioned at each end of the row?

9. How many 3 digit numbers can you create from the digits 1, 2, 3, 4, 5

(a) if digits can be repeated? (b) if digits cannot be repeated?


10

May 29, 2017

Calculating the Total Number of Possible Choices in a Counting Problem

Ø Can be attained by:

(i) Multiplication ‐ occurs when the word ‘AND’ is used. This indicates that these selections are performed together. (ie. Select from ‘a’ items AND from ‘b’ items AND from ‘c’ items…etc).

Number of possible choices = (a)(b)(c)…

(ii) Addition ‐ occurs when the word ‘OR’ is used. This indicates that ONLY ONE ITEM IS SELECTED from the entire group of all items combined. (ie. Select from ‘a’ items OR from ‘b’ items OR from ‘c’ items…etc). FCP does not apply here.

Number of possible choices = a + b + c + …


11

May 29, 2017

10. A school cafeteria offers a lunch menu.Determine the number of possible selections if:(a) choosing a soup and sandwich and drink and dessert.

(b)choosing a soup or sandwich or drink or dessert.


12

May 29, 2017

(IV) Factorials

11.How many ways can you arrange the letters in the word MEXICO?

Solution: The basic idea is we have 6 objects and 6 possible positions they can fill.

There is a shorter way of approaching this problem. It is called the Factorial Function and your calculator will perform this calculation for you! That is:

6! = 6 × 5 × 4 × 3 × 2 × 1 = 720


13

May 29, 2017

Factorial Notation> The product of descending consecutive natural numbers.

n! = n ×(n –1) ×(n –2) ×. . . ×3 ×2 ×1 where n is a natural number.

> What does 0! represent ?Answer: The number of ways to count the empty set.

0! = _____> n! represents the number of ways to arrange n distinct objects in a row

12. How many ways can you arrange five books on a shelf?

13. How many ways can you arrange 10 people in a row?


14

May 29, 2017

14. Evaluate the following expressions involving factorial notation.

15. Simplifying factorial expressions.


15

May 29, 2017

16. Solve:


16

May 29, 2017

17.


17

May 29, 2017

(V) Permutations – Arranging Objects where Order is important − Arranging a subset of items

Sometimes you will be given several items but you only want to arrange a few or ALL of them. This is referred to as a Permutation.

• A permutation is an ordered arrangement of all or part of a set.• For example, the possible permutations of the letters A, B and C are ABC, ACB, BAC, BCA, CAB and CBA.

• The order of the letters matters.

The number of permutations from a set of n different objects, where r of them are used in each arrangement, can be calculated using the formula:

The above formula implies NO REPETITIONS.· n is the total number of items · r is the number of items you want to permute or arrange.

nPr = where


18

May 29, 2017

18. How many ways can 8 students be selected for the positions of president, vice –president and secretary?

Fundamental Counting Principle:

Permutation Formula:


19

May 29, 2017

19. Evaluate using factorial notation.

(a) 6P2 (b)4P4


20

May 29, 2017

20. Simplify:


21

May 29, 2017

21. Solve for the variable.

(a) nP3 = 60 (b)n – 1P2 = 12


22

May 29, 2017

(c) 5Pr = 20 (d) 8Pr = 8!

23. 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?


23

May 29, 2017

Permutations

A. Arranging a Subset of Items: Sometimes you will be given a number of objects and you want to arrange only a few of them. (Use nPr or FCP.)

1. How many three‒digit numbers can be made using the numbers 1 through 9 if no numbers can be repeated?

2. How many 4 letter words can be created if repetitions are not allowed?

3. There are 7 members on the student council. How many ways can the council select 3 students to be president, vice president and secretary?

4. A code consists of 3 letters and 3 digits with no repetitions of digits or letters. Determine the number of possible codes.


24

May 29, 2017

Solving a Permutation Problem with Restrictions

• Permutation problems sometimes involve restrictions. In certain situations, objects may be arranged in a line where two or more objects must be placed together, or certain objects(s) must be placed in certain positions.

B. 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?


25

May 29, 2017

C. 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

1. How many arrangements of the word ACTIVE are there if C & E must always be together?

2. How any arrangements of the word Kitchen exist if K, I, and N must always be together?

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) the girls must stay together?

B) Brady and Dylan must stay together?

4. How many ways can 3 different math books, 5 different chemistry books, and 7 different physics books be arranged on a shelf if the books of each subject must be kept together?


26

May 29, 2017

D. Never Together: If certain items must be kept apart, you will need to figure out how many possible positions the separate items can occupy.

# ways not together = total # of arrangements # of arrangements together

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?

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?


27

May 29, 2017

Solving a Permutation Problem involving Cases

If a counting problem has one or more conditions that must be met,*Consider each case that each condition creates, and*Add the number of ways each case can occur to determine the total number of outcomes.*KEY WORDS: "at most", "at least" and “or”.

1. How many 2, 3 and 4 letter words can be formed from the word CANS?

Solution:

Case 1: Number of 2letter words



Total number of words:


28

May 29, 2017

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?

Case 1: Number of 4digit codes



Total number of codes:


29

May 29, 2017

3. How many different 4digit odd numbers greater than 300 can you make using the digits 1 – 7, if the numbers must be less than 6000? No digits are repeated.


30

May 29, 2017

1. A set of 6 marbles consists of 2 identical green marbles and 4 identical blue marbles. How many arrangements are possible?

2. How many ways can you arrange the letters in the word REFERENCE?

3. If there are 8 cookies (4 chocolate chip, 2 oatmeal, and 2 chocolate), in how many ways can you eat all of them?


31

May 29, 2017

4. Marion has to walk 6 blocks east and 4 blocks north. How many routes can she take?

5. How many different ways can you order the letters of Toronto if it begins with exactly two O's?

6. How many ways can the letters in CANADA be arranged if the first letter must be N and the last letter C ?


32

May 29, 2017

LAST MATH CHAPTER EVER - Weeblymsbourgeois2016.weebly.com/uploads/5/5/9/1/...Section 9.1...

Documents

Transcript of LAST MATH CHAPTER EVER - Weeblymsbourgeois2016.weebly.com/uploads/5/5/9/1/...Section 9.1...