Permutations and Combinations - Lecture Notes | MATH 1313, Study notes of Mathematics

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Section 6.4: Permutations and Combinations

Section 6.4: Permutations and Combinations

Definition: n-Factorial For any natural number n, ᡦ䙦ᡦ − 1䙧䙦ᡦ − 2䙧 … 3 ∙ 2 ∙ 1 0! = 1

A permutation is an arrangement of a specific set where the order in which the objects are arranged is important.

Formula: ᡂ䙦ᡦ, ᡰ䙧 = ぁ! 䙦ぁ⡹ぅ䙧! ,^ ᡰ ≤ ᡦ

where n is the number of distinct objects and r is the number of distinct objects taken r at a time.

Formula: Permutations of n objects, not all distinct

Given a set of n objects in which ᡦ⡩ objects are alike and of one kind, ᡦ⡰ objects are alike and of another kind,…, and, finally, ᡦぅ objects are alike and of yet another kind so that

ᡦ⡩ + ᡦ⡰ + ⋯ + ᡦぅ = ᡦ

then the number of permutations of these n objects taken n at a time is given by

ᡦ! ᡦ⡩! ᡦ⡰! … ᡦぅ!

A combination is an arrangement of a specific set where the order in which the objects are arranged is not important.

Formula: ᠩ䙦ᡦ, ᡰ䙧 = ぁ! ぅ!䙦ぁ⡹ぅ䙧! ,^ ᡰ ≤ ᡦ

where n is the number of distinct objects and r is the number of distinct objects taken r at a time.

Example 1: You are in charge of seating 5 honored guests at the head table of a conference. How many seating arrangements are possible if the 5 chairs are on one side of the head table?

Section 6.4: Permutations and Combinations

Example 7: In a production of West Side Story , eight actors are considered for the male roles of Tony, Riff, and Bernardo. In how many ways can the director cast the male roles?

Example 8: How many permutations can be formed from all the letters in the word MISSISSIPPI.

Example 9: A museum of fine arts owns 8 paintings by a given artist. Another fine arts museum wishes to borrow 3 of these paintings for a special show. How many ways can 3 paintings be selected for shipment out of the 8 available?

Example 10: A certain company has to transfer 4 of its 10 junior executives to a new location, how many ways can the 4 executives be chosen?

Example 11 : A student belongs to a entertainment club. This month he must purchase 2 DVDs and 3 CDs. If there are 10 DVDs and 10 CDs to choose from, in how many ways can he choose his 5 purchases?

Section 6.4: Permutations and Combinations

Example 12: A coin is tossed 5 times. a. In how many outcomes do exactly 3 heads occur?

{(H 1 H 2 H 3 TT), (H 1 H 2 T H 4 T), (H 1 H 2 TT H 5 ), (H 1 TH 3 T H 5 ), (H 1 TTH 4 H 5 ), (H 1 T H 3 H 4 T), (TH 2 H 3 H 4 T), (TH 2 H 3 T H 5 ), (TH 2 TH 4 H 5 ), (TTH 3 H 4 H 5 )}

b. In how many outcomes do at least 4 heads occur?

{(H 1 H 2 H 3 H 4 T), (H 1 H 2 H 3 T H 5 ), (H 1 H 2 TH 4 H 5 ), (H 1 TH 3 H 4 H 5 ), (TH 2 H 3 H 4 H 5 )}

{(H 1 H 2 H 3 H 4 H 5 )}

Example 13: A coin is tossed 20 times. a. In how many outcomes do exactly 7 tails occur?

b. In how many outcomes do at most 18 heads occur?

Example 14: A computer store receives a shipment of 35 laser printers, including 6 that are defective. Five of these printers are selected to be displayed in the store. a. How many of these selections will contain no defective printers?

b. How many of these selections will contain at least 1 defective printer?

Section 7.1: Experiments, Sample Spaces, and Events

Section 7.1: Experiments, Sample Spaces, and Events

An experiment is an activity with observable results (outcomes).

A sample point is an outcome of an experiment.

A sample space is a set consisting of all possible sample points of an experiment.

A Finite Sample Space is a sample space with finitely many outcomes.

An event is a subset of a sample space of an experiment.

Given two events, E and F: The union of E and F is denoted by E∪ F.

The intersection of E and F is denoted by E∩ F.

If E∩ F = Ø then E and F are called mutually exclusive. (An event is mutually exclusive also means that two events that cannot happen at the same time, such as getting a head and a tail on the same toss of a coin).

The complement of an event is Ec^ and is the set of all outcomes in a sample space that is not in E.

Example 1: Consider the experiment of tossing a die. a. Describe the sample space, S, of this experiment.

b. Let E be the event that an even number is tossed and F be the event that a prime number is tossed. Describe E and F in set notation then find the following:

E∪ F =

E∩ F =

Ec=

(E∪ Fcc^ )c^ =

Section 7.1: Experiments, Sample Spaces, and Events

Example 2: A sample of 3 apples taken from a fruit stand is examined to determine whether they are good or rotten. The sample space S = {GGG, GGR, GRG, GRR, RGG, RGR, RRG, RRR}. Let E be the event that at least 2 apples are good and let F be the event that exactly 2 apples are rotten. Find the event.

Example 3: An experiment consists of selecting a letter at random from the letters in the word DALLAS and observing the outcomes. a. What is an appropriate sample space for this experiment?

b. Describe the event “the letter selected is a vowel.”

Example 4: Describe a sample space associated with the experiment of tossing 2 fair coins.

Describe the event of having the same outcome on each coin.

Section 7.2: Definition of Probability

A sample space in which the outcomes of an experiment are equally likely to occur is called a uniform sample space . Let S={s 1 , s 2 ,…, s n } be a uniform sample space. Then

ᡂ䙦s⡩䙧 = ᡂ䙦s⡰䙧 = ⋯ = ᡂ䙦sぁ䙧 =

Finding the probability of an Event E:

1. Determine the sample space S.
2. Assign probabilities to each of the simple events of S.
3. If E={ s 1 , s 2 ,…, s k } where {s 1 }, {s 2 },…, {sk} are simple events then

ᡂ䙦ᠱ䙧 = ᡂ䙦ᡱ⡩䙧 + ᡂ䙦s⡰䙧 + ⋯ + ᡂ䙦s〸䙧

Note: If E = Ø then P(E) = 0.

Example 3: A pair of fair dice is cast. What is the probability that

a. the sum of the numbers shown is less than 5?

b. at least one 6 is cast?

c. you roll doubles?

Section 7.2: Definition of Probability

Example 4: If one card is drawn from a well-shuffled standard 52-card deck, what is the probability that the card drawn is a. A club?

b. A red card?

c. A seven?

d. A face card?

e. A black 9?

Popper 3: Suppose you buy a piece of office equipment for $19,000.00. After 5 years you sell it for

a scrap value of $5,000.00. The equipment is depreciated linearly over 5 years. The rate of

depreciation for the piece of equipment is

Popper 4: Find the inverse of the given matrix, if it exists.

a. 䙲^6 − −7 8

b. 䚄

⡴ ⡩⡱ −^

⡵ ⡩⡱ − ⡳ ⡩⡱ −^

⡶ ⡩⡱

c. 䚄

⡴ ⡩⡱

⡵ ⡩⡱ ⡳ ⡩⡱

⡶ ⡩⡱

d. No inverse exist

Popper 5: A problem is listed below. Identify its type.

Beginning in 2 years, a new Booster Club at a certain high school would like to award an outstanding senior on the varsity soccer team with a check of $4,000 towards the student's college tuition. Since there is a boy and girl team, they wish to award one senior boy and one senior girl the same amount of money. Thus, the club will need $8,000 at the end of 2 years. How much must the booster club invest each semiannual period in an account that pays 4% per year compounded semiannually to have the desired funds in 2 years

a. Amortization b. Future Value of an Annuity c. Present Value of an Annuity d. Sinking Fund

Popper 6: Charles will supplement his diet by adding at least 76 mg of Vitamin B and at least 52

mg of Vitamin C per day. He can get the desired amount of vitamins from two brands of vitamins -

Feel Great and Build Up. Each pill of Feel Great contains 10 mg of Vitamin B and 5 mg of Vitamin C.

Each pill of Build Up contains 8 mg of Vitamin B and 9 mg of Vitamin C. Each pill of Feel Great costs $1.12 and each pill of Build Up costs $1.15. Suppose you want to know how many pills per day of

each brand Charles should take to minimize cost. Let x = the number of pills of Feel Great and y =

the number of pills of Build Up. Which of the following, if any, would be an appropriate objective

function?

a. Minimize ᠩ = 1.12ᡶ + 9ᡷ b. Minimize ᠩ = 1.12ᡶ + 1.15ᡷ c. Minimize ᠩ = 1.15ᡶ + 5ᡷ d. Minimize ᠩ = 1.15ᡶ + 1.12ᡷ

Popper 7: Given that the augmented matrix in row-reduced form below is equivalent to the augmented matrix of a system of linear equations. Determine whether the system has a solution

and find the solution(s) to the system, if they exist

a. ᡶ = 7, ᡷ = 10 b. ᡶ = −7, ᡷ = −10, ᡸ = 6 c. ᡶ = −7, ᡷ = − d. No Solution

Popper 8: Give the optimal solution (point and value).

Maximize : ᡂ = 5ᡶ + 8ᡷ ᡶ + ᡷ ≤ 9 ᡶ + 2ᡷ ≤ 10 ᡶ ≥ 0, ᡷ ≥ 0

a. ᡂ = 50 ᡓᡲ (10, 0) b. ᡂ = 72 ᡓᡲ (0, 9) c. ᡂ = 48 ᡓᡲ (8, 1) d. ᡂ = 40 ᡓᡲ (0, 5)