# Counting Principles (Permutations and Combinations )

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• Counting Principles(Permutations and Combinations )

• Example A combination lock can be set to open to any 4-digitsequence. (a) How many sequences are possible? (b) How many sequences are possible if no digit is repeated?

2012 Pearson Education, Inc.. All rights reserved. Solution: (a) Since there are 10 digits namely 0, 1, 2..9, thereare 10 choices for each of the digit. By the multiplicationprinciple, there are 10 10 10 10 =10,000 different sequences.

(b)There are 10 choices for the first digit. It cannot be usedagain, so there are 9 choices for the second digit, 8 choices forthe third digit, and then 7 choices for the fourth digit.Consequently, the number of such sequences is 10 9 8 7 =5040 different sequences.

• 2012 Pearson Education, Inc.. All rights reserved. ExampleHow many different ways can you choose a bagel, muffin or donut to eat and coffee or juice to drink?Tree diagram

• Example A teacher is lining up 8 students for a spelling bee. How manydifferent line-ups are possible? 2012 Pearson Education, Inc.. All rights reserved. Solution: Eight choices will be made, one for each space that willhold a student. Any of the students could be chosen for the first space. There are 7 choices for the second space, since 1 studenthas already been placed in the first space; there are 6 choices forthe third space, and so on. By the multiplication principle, the number of different possiblearrangements is 8 7 6 5 4 3 2 1 = 40,320.

• Example A teacher wishes to place 5 out of 8 different books on hershelf. How many arrangements of 5 books are possible? 2012 Pearson Education, Inc.. All rights reserved. Solution : The teacher has 8 ways to fill the first space, 7 waysto fill the second space, 6 ways to fill the third, and so onSince the teacher wants to use only 5 books, only 5 spaces canbe filled (5 events) instead of 8, for 8 7 6 5 4 = 6720arrangements.

• Example Find the number of permutations of the letters L, M, N, O, P,and Q, if just three of the letters are to be used.Solution: 2012 Pearson Education, Inc.. All rights reserved.

• 2012 Pearson Education, Inc.. All rights reserved. How many permutations are there of two letters from the set {A,B,C}.

• Example How many committees of 4 people can be formedfrom a group of 10 people?Solution: A committee is an unordered group, so use thecombinations formula for C(10,4).

• Example From a class of 15 students, a group of 3 or 4 students will beselected to work on a special project. In how many ways can agroup of 3 or 4 students be selected?Solution: The number of ways to select group of 3 studentsfrom a class of 15 students is C(15, 3) = 455.The number of ways to select group of 4 students from a classof 15 students is C(15, 4) = 1365.The total number of ways to select a group of 3 or 4 studentswill be the 1820. 2012 Pearson Education, Inc.. All rights reserved.

• Example (a) How many 4-digit code numbers are possible if no digits are repeated?Solution: Since changing the order of the 4 digits results in adifferent code, permutations should be used.

(b) A sample of 3 light bulbs is randomly selected from a batch of 15. How many different samples are possible?

Solution: The order in which the 3 light bulbs are selected is notimportant. The sample is unchanged if the items are rearranged,so combinations should be used.

• Example

(c) In a baseball conference with 8 teams, how many games must be played so that each team plays every other team exactly once?

Solution: Selection of 2 teams for a game is an unorderedsubset of 2 from the set of 8 teams. Use combinations again. (d) In how many ways can 4 patients be assigned to 6 differenthospital rooms so that each patient has a private room?

Solution: The room assignments are an ordered selection of 4rooms from the 6 rooms. Exchanging the rooms of any 2patients within a selection of 4 rooms gives a differentassignment, so permutations should be used. 2012 Pearson Education, Inc.. All rights reserved.

• Introduction to Probability

• Write the elements belonging to the set {x | x is a state whosename begins with the letter O}.

2012 Pearson Education, Inc.. All rights reserved. Solution: The state names that begin with the letter O make upthe set {Ohio, Oklahoma, Oregon }.Set a collection of elements that satisfy a certain condition.

• 2012 Pearson Education, Inc.. All rights reserved. Example 1 Decide if the statement is true or false.

Solution: The first set is a subset of the second because each element of first set belongs to second set. (The fact that the elements are listed in a different order does not matter.) Therefore, the statement is true.

• 2012 Pearson Education, Inc.. All rights reserved. A) What is the sample space?B) What is the probability of getting a 3?C) What is the probability of getting an odd number?Sample Space The set of all possible outcomes of an experiment or event.

• ExampleTwo coins are tossed, and a head or a tail is recorded for eachcoin. Write the event E: the coins show exactly one head.

Solution: Tossing a coin is made up of the outcomes heads (H)or tails (T). If S represents the sample space of tossing twocoins, then S = {HH, HT, TH, TT}.Two outcomes satisfy this condition: so,E = {HT, TH}.

• ExampleTwo coins are tossed, and a head or a tail is recorded for eachcoin. Give a sample space for this experiment.What is the probability of getting at least one head.Solution: A) Tossing a coin is made up of the outcomes heads (H)or tails (T). If S represents the sample space of tossing twocoins then S = {HH, HT, TH, TT}. Solution : B) E = {HH,HT,TH} 2012 Pearson Education, Inc.. All rights reserved. Number of elements in the sets

• http://www.youtube.com/watch?v=mhlc7peGlGg&safe=active Monty Hall probability puzzle

• Probability of Multiple Events

• 2012 Pearson Education, Inc.. All rights reserved. 46512(no outcomes in common; if one event occurs the other cannot)(E is called the complement of E)

• Example

A study of workers earning the minimum wage grouped such workers into various categories, which can be interpreted as events when a worker is selected at random. Source:Economic Policy Institute. Consider the following events:

E: worker is under 20;F: worker is white;G: worker is female.

• 2012 Pearson Education, Inc.. All rights reserved. Independent Events The fact that one event has occurred, does not change the probability of the second event occurring.** Events that are mutually exclusive must be dependent, but dependent events are not necessarily mutually exclusive**Read as; the probability of event F occurring given E has occurredRead as; the probability of event E occurring given F has occurred

• Example If a single card is drawn from an ordinary deck of cards, findthe probability of an ace or a club.

Solution: Let A represent the event an ace and C the eventclub card. There are 4 aces in the deck, so

There are 13 clubs in the deck, so

Since there is 1 ace of club in the deck, By the union rule, the probability of the card being an ace or aClub card is 2012 Pearson Education, Inc.. All rights reserved.

• ExampleSuppose two fair dice are rolled. Find the probability that the sum is 8, or both die show the same number.Solution:

• ExampleFind the probability that when two fair dice are rolled, the sumis less than 11.

Solution: To calculate this probability directly, we must find theprobabilities that the sum is 2, 3, 4, 5, 6, 7, 8, 9 or 10 and thenadd them. It is much simpler to first find the probability of the complement, the event that the sum is greater than or equal to 11.

• (General rule of multiplication)When events are Independent;

So the equation simplifies to:

• IndependentNot IndependentIndependentNot Independent1/60.542/79/349/25Not mutually exclusivemutually exclusiveNot mutually exclusive

• 2012 Pearson Education, Inc.. All rights reserved. P(F); represented by large circle.P(E and F); represented by overlapping section

• Example

The following table shows national employment statistics. Use the table to find each probability.8937580143839392972919121

7.P(male | professional)8.P(laborer | female)9.P(female | sales)10.P(professional | female)11.P(sales | male)12.P(male | laborer)

• ExampleUse a tree diagram to solve the following problem.

16.A car insurance company compiled the following information from

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