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Intro to

Probability and Statistics

Unit 2

Permutations, Combinations, Binomial Theorem

Table of Contents

I. Counting Method ………………………….pg 4

II. Permutations ………………………………pg 8

III. Combinations………………………………pg 12

IV. Probability Involving …………………….. pg 16

Permutations and Combinations

V. Binomial Expansions………………………pg 20

VI. Probabilities Involving ……………………pg 25

the Binomial Theorem

Vocabulary (Words, phrases and formulas you need to know)

1. Permutation

2. n factorial, n!

3. Permutations of n objects

4. Permutations of n objects taken r at a time

5. Circular Permutations

6. Permutations with identical objects

7. Combination

8. Combinations of n objects taken r at a time

9. What is the difference between a permutation and a combination?

CHOICES….CHOICES….CHOICES….

1. 3 coins are flipped. List all of the possible ways the 3 coins can land. For example: one possible way is Heads-Tails-Heads (HTH).

2. You are going to make a peanut butter and jelly sandwich for lunch. You can first choose either white or wheat bread. Then you can choose from either chunky or smooth peanut butter. Finally, you can choose a jelly made from grapes, apples or raspberries. List all the different sandwiches that can be made. For example, one choice is Wheat-Chunky-Apple (WheatCA).

3. Five kindergarden students are going to the art room with their teacher. They are Jimmy, Sarah, Karen, AJ and Derek. One of the students will be the line leader, while a different student will be in charge of closing the door behind them. How many different ways can we assign the jobs? For example, one way to assign the jobs is Sarah - line leader, and Derek - door duty.

4. As part of a survey, Philadelphia sports fans are asked to list the 4 pro sports teams, from favorite to least favorite. In how any ways can someone answer the survey? For example, one way is Eagles-Phillies-Sixers-Flyers.

5. How many different possible 7-digit phone numbers exist? For example, one possibility is 659-0135. What restrictions are there on phone numbers? Are there phone numbers that are not allowed? Think of and list these restrictions.

FACTORIAL

The expression n! means .

For example, 6! means .

We can simplify expressions involving factorial by .

Ex: [pic]

Ex: [pic]

Simplify each of the following:

1. 8! =

2. 10! =

3. [pic]

4. [pic]

5. [pic]

6. [pic]

7. [pic]

8. [pic]

Notes:

Ex 1: Pizzas at Mr. L’s pizza shop come in 3 sizes, with 3 types of crusts (reg., pan, stuffed), and with or without cheese. How many different types of pizzas can be ordered?

______________________

The COUNTING PRINCIPLE says we can multiply the number of each choice to get the total number of choices.

Example 2: At a restaurant at Center City, you have the choice of 8 different entrees, 2 different salads, 12 different drinks, & 6 different deserts.

How many different dinners (one choice of each) can you choose?

______________________

Fund. Counting Principle with and without repetition

Example 3: Ohio Licenses plates have 3 #’s followed by 3 letters.

A. How many different licenses plates are possible if digits and letters can be repeated?

______________________

B. How many plates are possible if digits and numbers cannot be repeated?

______________________

Practice Set 1:

1. How many three-digit numerals can be written using the symbols 6, 7, and 8?

2. How many different ways can ten questions on a true-false test be answered?

3. An automobile manufacturer produces 7 models, each available in 6 different colors. In addition, the buyer can choose one of 4 different upholstery fabrics and one of 5 different colors for the interior. How many varieties of cars can be ordered from the manufacturer?

4. A witness to a holdup reports that the license of the getaway car consisted of 6 different digits. He remembers the first three but has forgotten the remainder. How many licenses do the police have to check?

5. How many three-letter arrangements can be made of the letters P, R, I, M, and E, if any letter may be repeated?

6. How many positive odd integers whose numerals contain three digits can be formed, using the digits 1,2,3,4, and 5? (Hint: Fill the units place with an odd digit; then fill the remaining places.)

7. How many positive even integers of three digits can be formed from the digits 1,2,3,4, 5, and 6? (Hint: Fill the units place with an even digit; then fill the remaining places.)

8. How many auto license plates of four symbols can be made in which at least two of the symbols are letters and the rest are digits?

Example 1:

In how many ways can 8 students get in line for the bus?

______________________

Each arrangement of items in a particular order is called a PERMUTATION.

The number of ways they can get in line is 8x7x6x5x4x3x2x1 = 8!

The number of permutations of n items is given by: n !

Example 2:

[pic]

Example 3:

How many different ways can 12 skiers in the Olympic finals finish the competition? (if there are no ties)

______________________

Example 4:

There are 12 horses in a race. In how many ways can the horses finish in 1st, 2nd and 3rd place?

______________________

[pic]

Example 5: You are choosing between 10 colleges, you want to visit all or some.

A. How many ways can you visit 6 of them?

______________________

B. How many ways can you visit all 10 of them?

______________________

[pic]

[pic]

[pic]

Example 6:

Find the number of distinguishable permutations of the letters:

• SUMMER : ______________________

• WATERFALL : ______________________

Example 7:

A dog has 8 puppies, 3 male and 5 female. How many birth orders are possible?

______________________

[pic][pic]

Practice Set 2:

1. In how many ways can the letters in the word PHOENIX be arranged if each letter is used only once in each arrangement?

2. Seven salesmen are to be assigned to seven different counters in a department store. How many ways can the assignment be made?

3. A school has six sections of first-year algebra. In how many ways can a pair of twins be assigned to algebra classes if their parents have requested that they be placed in different classes?

4. A business school gives courses in typing, shorthand, transcription, business English, technical writing, and accounting. In how many ways may a student arrange his program if he has three courses a day?

5. To lead a certain cheer, the seven cheerleaders form a circle, each facing the center. In how many orders can they arrange themselves?

6. A milliner wants to arrange six different flowers around the brim of a hat. In how many orders can she place them?

7. In how many ways can five keys be arranged on a key ring?

Practice Set 3:

1. [pic] ________________ 2. [pic]________________

3. [pic] ________________ 4. [pic]________________

Find the number of permutations of the letters in each word.

5. geometry ________________ 6. algebra________________

7. calculus ________________ 8. mathematics________________

9. Lizette decorates windows for a department store. She plans to design a baby’s room with a row of stuffed elephants and monkeys along one wall. If she has 8 identical elephants and 10 identical monkeys, in how many different ways can the stuffed animals be displayed?

10. The 6 candidates for a student government office are invited to speak at an election forum. In how many different orders can they speak?

11. Representatives from 8 schools are represented at a school newspaper workshop. In how many different ways can the 8 representatives be seated around a circular table?

12. Ten colleges are participating in a college fair. Booths will be positioned along one wall of a high school gymnasium. In how many different orders can the booths be arranged?

In the last section we learned counting problems where order was important

For other counting problems where order is NOT important like cards, (the order you’re dealt is not important, after you get them, reordering them doesn’t change your hand)

These unordered groupings are called Combinations. A Combination is a selection of r objects from a group of n objects where order is not important

[pic]

[pic]

[pic]

[pic]

[pic]

Practice Set 4:

1. How many combinations can be formed from the letters D R E A M, taking two at a time?

2. In how many ways can a student choose to answer five questions out of eight on an examination, if the order of his answers is of no importance?

3. Eva's Hamburger Haven sells hamburgers with cheese, lettuce, tomato; relish, ketchup, or mustard. How many different hamburgers can be made, choosing any three of the "extras"?

4. Julie owes letters to her grandmother, her uncle, her cousin, and two school friends. Tuesday night she decides to write to two of them. From how many combinations can she choose?

5. Local 352 is holding an election of four officers. In how many ways can they be chosen from a membership of 75? (Disregard order.)

6. The twelve engineers in the instrumentation department of Ajax Missile Corp. are to divide into project groups of four persons each. How many possible groups are there?

7. How many different five-card hands can be drawn from a pack of 52?

[pic]

When finding the number of ways both an event A and an event B can occur, you multiply.

When finding the number of ways that an event A OR B can occur, you add.

[pic]

[pic]

Practice Set 5:

Find the number of ways in which each committee can be selected.

1. A committee of 5 people from a group of 8 people.

2. A committee of 4 people from a group of 7 people.

3. A committee of 8 people from a group of 15 people.

At a luncheon, guests are offered a selection of 4 different grilled vegetables and 5 different relishes. In how many ways can the following items be chosen?

4. 2 vegetables and 3 relishes

5. 3 vegetables and 2 relishes

A bag contains 8 white marbles and 7 blue marbles. Find the probability of selecting each combination.

6. 2 white and 3 blue

7. 3 white and 2 blue

Determine whether each situation involves a permutation or a combination. Do not solve!!

8. A high school offers 5 foreign language programs. In how many ways can a student choose 2 programs?

9. In how many ways can 20 members be chosen from a 60 member chorus to sing the national anthem at a graduation ceremony?

10. In how many ways can a captain, co-captain, and team manager be chosen from among the 18 members of a volleyball team?

Practice Set 6:

1. [pic] ________________ 2. [pic]________________

3. [pic] ________________ 4. [pic]________________

5. How many different committees of four can you form from 10 people?

6. How many different five-person work crews can you form from 18 people?

7. There are eight outfits you would like to buy for your weekend job, but you only saved enough money for three. How many choices do you have?

8. A high school offers 5 foreign language programs. In how many ways can a student choose 2 of the programs?

9. In how many ways can 6 students from a 10-member chorus be chosen to sing the national anthem at a baseball game?

10. In how many ways can a captain, co-captain and a team manager be selected from a 18-member volleyball team?

11. First through fourth place prizes are to be awarded in a poetry contest. In how many ways can the prizes be awarded if there are 20 entries?

12. How many different 5-card hands can be dealt from a standard deck of 52 cards?

13. There are 8 essay questions on your social studies final exam. You are told to select 5 of the essay questions to answer. In how many ways can this be done?

14. In how many ways can Mrs. Cassidy arrange the 12 Care Bears she keeps on a shelf in her den?

15. In how many ways can Mr. L selct 3 of his 25 students to go outside and clap erasers?

PROBABILITIES INVOLVING COMBINATIONS

Problems 1-4 refer to the following experiment: 5 cards are drawn from a deck. Find the probability that the cards drawn include the cards listed. First, write your answer using P or C notation. Then compute each answer as a percentage to 3 significant digits.

1. 3 spades and 2 hearts

2. 5 spades

3. 2 kings and 3 queens

4. at least 3 fours

Problems 5 to 8 refer to the following experiment: 3 balls are drawn from an urn containing 20 red balls, 15 green balls, and 10 white balls. Find the probability that the balls drawn are as described. First, write your answer using P or C notation. Then compute each answer as a percentage to 3 significant digits.

5) 3 green balls

6) 2 green balls and 1 white ball

7) 2 red balls and 1 white ball

8) at least 1 red ball

The FBLA is going to send 6 students to a national convention. To be fair, they will choose names at random to decide who gets to go. Find the probability that the following groups of students are chosen, using their enrollment chart below as a guide.

|SEX \ GRADE |9th |10th |11th |12th |

|Male |3 |4 |8 |10 |

|Female |7 |6 |7 |5 |

9) 3 boys and 3 girls are chosen

10) 3 - 11th graders and 3 - 12th graders are chosen

11) 5 girls and one boy (lucky guy!) are chosen

12) No 9th graders are chosen

Practice Set 8:

1. There are 12 fish in a tank, 5 are pufferfish, 4 are monkfish and 3 are goldfish. If you select three fish, what is the probability that you select one of each type?

2. Using a normal deck of cards, you select 7 cards, what is the probability of getting exactly 3 queens?

3. There are 7 girls and 5 boys on the math team. If 4 people are chosen randomly from the team, what is the probability of selecting 2 boys and 2 girls?

4. A package of 55 apples contains 5 that are bruised. If you select 12 apples, what is the probability that 3 of the ones you select are bruised?

5. From a normal deck of cards, 3 cards are selected and placed face up in order on the table. What is the probability that two of the cards are aces?

6. From a club containing 14 girls and 4 boys, 3 people are picked to go on the England Exchange. What is the probability that at least 2 girls are picked?

7. From a normal deck, you draw 7 cards. What is the probability you don’t draw a face card?

8. You draw two marbles from a bag of 5 green, 4 red, and 2 blue marbles. What is the probability that you draw a blue then a green?

9. 1st through 4th place are awarded in an art contest. In how many ways can winners be selected from the 115 entries?

10. A shipment of 50 X-boxes contains 10 that are defective. What is the probability that out of 15 X-boxes chosen randomly, 9 of them are defective?

11. A bag contains 12 marbles. 3 are red, 4 are blue, 2 are green, and 3 are yellow. If you pick 4 marbles, what is the probability of pulling one of each color, in rainbow order? (Order of the rainbow is red, orange, yellow, green, blue, violet)

12. Out of a normal deck of cards, what is the probability that a 7 card hand drawn at random will contain at least 4 clubs?

13. Ten paintings are submitted for an art contest. You painted 3 of them. 1st, 2nd and 3rd place prizes were awarded. What is the probability that you win all three prizes?

14. In student council, there are 92 members. 36 of them are upperclassmen. What is the probability that out of 12 random members, 5 are upperclassmen and the rest are not?

[pic]

Example Problems:

Use the Binomial Theorem to expand each binomial.

1. [pic]

2. [pic]

3. [pic]

4. [pic]

5. [pic]

[pic]

Use the Binomial Theorem to expand each binomial.

1. [pic]

2. [pic]

3. [pic]

4. [pic]

5. [pic]

6. [pic]

7. [pic]

8. [pic]

[pic]

[pic]

[pic]

[pic]

[pic]

[pic]

Practice Set 10:

For the expansion of [pic], find the indicated term.

1. 3rd term _______________________________________________

2. 6th term _______________________________________________

3. 10th term _______________________________________________

For the expansion of [pic], find the term that contains each monomial.

4. [pic] _______________________________________________

5. [pic] _______________________________________________

6. [pic] _______________________________________________

Expand each binomial.

7. [pic] _______________________________________________

8. [pic] _______________________________________________

9. [pic] _______________________________________________

|Expand: |Find the 6th term of |Expand: |

| | | |

|[pic] |[pic] |[pic] |

|Expand: |Find the 7th term of |Expand: |

| | | |

|[pic] |[pic] |[pic] |

|Expand: |Find the 4th term of |Expand: |

| | | |

|[pic] |[pic] |[pic] |

You must complete 3 in a row, either diagonally, horizontally or vertically.

Probability Problems using the Binomial Theorem

Example 1:

Mark Price has held a NBA career record for free throws. He attempted more than 2000 free throws and made 90% of them. If his probability of success on any single shot is 0.9, find the probability that he will make exactly 6 out of 10 consecutive shots.

Find the probability that he will make exactly 9 out of 10 shots.

Example 2:

Ken Griffey’s batting average is .300. What is the probability of him getting exactly 3 hits in his next 5 at bats?

What is the probability of him getting at least 2 hits in his next 5 at bats?

What is the probability of him getting 3 or more hits in his next 5 at bats?

PROBABILITY PROBLEMS INVOLVING THE BINOMIAL THEOREM

1. While Shaquille O’Neil knows how to dunk a basketball, he is not a great free throw shooter. This past season, Shaq made about 55% of his free throws. In a game against Seattle, Shaq took 8 foul shots.

What is the probability that he made EXACTLY 6 of these foul shots?

What is the probability that he made EXACTLY 7 of these foul shots?

What is the probability that he made AT LEAST 6 of these foul shots?

2. A hatching yellow-bellied sapsucker has a 0.58 probability of surviving to adulthood. I have a nest of 6 eggs in my tree. What is the probability that all 6 hatchlings will survive to adulthood?

What is the probability that 2 or less of them will survive?

3. Jack Brickowski plays basketball for the Juneau Polar Bears. In the past season, Jack made exactly 50% of his free-throws. What is the probability that:

Out of 10 free throws, he makes exactly 6.

Out of 10 free throws, he makes all 10.

Out of 6 free throws, he misses all 6.

4. Dr. Zeus is using a method in treating his patients that is 92% effective.

What is the probability that there will be no failures in 30 treatments?

What is the probability that there will be 3 failures in 30 treatments?

What is the probability that there will be less than 3 failures in 30 treatments?

5. The weather forecast is for a 20% chance of rain during each of the next four days. What is probability that it will rain at least one day of the four?

6. What is the probability that you will roll exactly five sixes in ten tosses of a number cube?

7. You friend’s batting average is 0.225. What is the probability of her getting three or more hits in the next five times at bat?

8. The weather forcast says a 75% chance of rain each day for the next five days. What is the probability that will rain two of the next five days?

What is the probability that it will rain four of those days?

9. Jimmy’s batting average is .146. What is the probability that Jimmy will get four hits out of his next six times at bat?

What is the probability of Jimmy getting two hits out of his next four times at bat?

10. Samantha’s free throw average is 82%. What is the probability that Samantha will make at least three free throws out of five?

11. The probablity of buying a new car with engine problems is .002. What is the probability that “Rent-for-You” auto company can purchase five new cars without any engine problems?

|Matt is batting .427. What is the |The probability of the new school bell being|If the probability of the Eagles winning a |

|probability he gets exactly 5 hits in his |changed each day is 1%. Find the |game is 61%, find the probability they will |

|next 8 at bats? |probability that the bell will be changed |win at least 3 of the next 4 games. |

| |once a day for the next 4 out of 5 school | |

| |days. | |

|The likelihood of rain this week is 22% each|Donovan McNabb has a 59% completion rate. |Find the probability of getting 5 or more |

|day, find the probability it will rain no |Find the likelihood that he will complete |heads in 10 flips of a coin? |

|more than 3 of the next 7 days. |exactly 8 of his next 10 throws. | |

|Chase Utley got 7 hits in his last 12 at |Chase Utley got 7 hits in his last 12 at |The probability of the cafeteria offering a |

|bats, assuming he maintains that same |bats, assuming he maintains that same |deli bar on any given day is 28%. Find the |

|batting average, what is the probability he |batting average, what is the probability he |probability that a deli bar is offered for |

|will get exactly 5 hits in his next 10 at |will get at least 3 hits in his next 5 at |no more than 2 of the next 7 days. |

|bats? |bats. | |

You must complete 3 in a row, either diagonally, horizontally, or vertically.

Review Problems:

There is a 3% chance that the computer you ordered from Gateway will not work when it arrives.

1. What is the probability that all 10 of the computers Mr. Williams has ordered for the

High School will work.

2. What is the probability that at least 9 of the 10 will work?

Chipper Jones has a .310 batting average.

3. What is the probability that he will get 3 hits in his next 5 at bats?

4. What is the probability that he will get 1 or fewer hits in his next 5 at bats?

5. Expand [pic]

6. Expand [pic]

7. Expand [pic]

8. How many ways can Mr. Lochel pick 4 members of his math club to compete in a contest? There are 20 members of the club.

9. How many ways can student council pick 1st, 2nd, and 3rd places in the dance contest if there are 18 contestants?

10. How many ways can 5 friends stand in line at a movie theater?

11. How many ways can 6 recipes be chosen from a list of 35 to publish in a cookbook?

12. How many ways can student council elect a President, Vice-President and Treasurer from their 33 students?

Practice Test:

1. The probability of the new school bell being changed each day is 1%. Find the probability that the bell will be changed once a day for the next 4 out of 5 school days.

2. Expand : [pic]

3. Mr. Evans’ class has 24 students. How many groups of 4 could be formed?

4. Find the middle term of the expansion : [pic]

5. A cookie jar has 18 cookies: 6 chocolate chip, 5 sugar and 7 oatmeal raisin. If 7 cookies are drawn, what is the probability that exactly 3 of them are chocolate chip?

6. A bag contains 5 volleyballs, 4 basketballs and 6 soccer balls. 3 balls are randomly drawn. What is the probability at least 2 of them are volleyballs?

7. How many arrangements can be formed by the letters in the word : PHOTOGRAPH ?

8. Write the first three terms of [pic]

9. Matt is batting .427. What is the probability he gets exactly 5 hits in his next 8 at bats?

10. There are two bruised apples in an eleven-apple bag. What is the probability that the two apples you choose for lunch aren’t bruised?

11. Find the probability of getting 5 or more heads in 10 flips of a coin?

12. How many different ways are there to purchase 2 CD’s, 2 cassettes and 3 videotapes from a collection consisting of 10 CD’s, 12 cassettes and 9 videotapes?

-----------------------

3 x 3 x 2 =

18 different pizzas

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