Probability and statisticsM 381

Counting Rules

Counting Principles

The Multiplication PrinciplePermutationsCombinationsArrangement

Objectives

apply fundamental counting principle

compute permutations

compute combinations

distinguish permutations vs combinations

Warm UP

Alice can’t decide what to wear between a short, a pant, and a skirt. She has four tops that will go with all three pieces: one red, one black, one white, and one striped.How many different outfits could Alice form

from these items of clothing?

Bottom Top Outfit

Short

Pant

Skirt

RedBlackWhiteStriped

Red BlackWhiteStriped

RedBlackWhiteStriped

Short, Red Top

Short, Black TopShort, White Top

Short, Striped Top

If the tree diagram is finished, how many outfits will she have?

12 outfits!!

• Tree diagrams are not often convenient, or practical, to use when determining the number of outcomes that are possible.

• Rather than using a tree diagram to find the number of outfits that Alice had to choose from, we could have used a general principle of counting: the multiplication principle.

Multiplication Principle or The Basic Counting Rule

• Ex1: A product can be shipped by four airlines and each airline can ship via three different routes. How many distinct ways exist to ship the product?

• How many different license plates can be made if each license plate is to consist of three letters followed by three digits and repetition is allowed? ___ • ___ • ___ • ___ • ___ • ___ L L L D D D

• If repetition is not allowed?

26

26 26 10 10

10 = 26 ³ • 10 ³

= 17, 576, 000

___ • ___ • ___ • ___ • ___ • ___ L L L D D D

26 25 24 10 9 8 = 11, 232, 000

Ex2: How many different license plates can be made if each license plate begins with three distinct letters followed by two digits?

Ex3: How many different security 4-digit numbers are possible if the first digit may not be zero?

Ex4: Marie is planning her schedule for next semester. She has to take the following five courses: English, history, geology, psychology, and mathematics.

a.) In how many different ways can Marie arrange her schedule of courses?

b.) How many of these schedules have mathematics listed first?

Ex5: You are given the set of digits {1, 3, 4, 5, 6}.

a.) How many three-digit numbers can be formed?

b.) How many three-digits numbers can be formed if the number must be even?

c.) How many three-digits numbers can be formed if the number must be even and no repetition of digits is allowed?d.) How many three distinct digits numbers can be formed if 6 is one of the digits?

The number of ways to arrange the letters ABC:

____ ____ ____

Number of choices for first blank? 3 ____ ____

3 2 ___Number of choices for second blank?

Number of choices for third blank? 3 2 1

3*2*1 = 6 3! = 3*2*1 = 6

ABC ACB BAC BCA CAB CBA

Permutations

• A permutation of r (where r ≥ 1) elements from a set of n elements is any specific ordering or arrangement, without repetition, of the r elements.

• Each rearrangement of the r elements is a different permutation.

• Permutations are denoted by nPr or P(n, r)

Ex6: A chairperson and vice-chairperson are to be selected from a group of nine eligible people. In how many ways can this be done?

Factorial Notation

The Special Permutations Rule

The Special Permutations Rule: The number of possible permutations of m objects among themselves is m!Ex 7: A student has 10 books to arrange on a shelf of a bookcase. In how many ways can the 10 books be arranged?

Distinguishable Permutations

• If the n objects in a permutation are not all distinguishable – that is, if there so many of type 1, so many of type 2, and so on for r different types, then the number of distinguishable permutations is

n! . n ! n ! ••• n !1 r2

How many distinct arrangements can be formed from all the letters of SHELTONSTATE?

Step 1: Count the number of letters in the word, including repeats.

12 letters

Step 2: Count the number of repetitious letters and the number of times each letter repeats.

S : 2 repeats E : 2 repeats T : 3 repeats

Solution: 12! . 2! 2! 3!

= 19, 958, 000

• Ex8:In how many distinct ways can the letters of MATHEMATICS be arranged?

Combination

Suppose you want to buy three CD’s from a selection of five CD’s. There are 10 ways to make your selections

ABC,ABD, ABE, ACD, ACE, ADE, BCD, BCE, BDE, CDE.In each selection, order does NOT matter. (ABC is

the same set as BAC). The number of ways to choose r objects from n objects without regard to order is called the number of combinations of n objects taken r at a time.

A combination is a selection of r objects from a group of n objects without regard to order and is denoted by n C r. The number of combinations of r objects selected from a group of n objects is:

!)!(

!

rrn

nCrn

Since the order does not matter in combinations, there are fewer combinations than permutations.  The combinations are a "subset" of the permutations.

• Ex9: To play a particular card game, each player draw five cards from a standard deck of 52 cards. How many different hands are possible?

• Ex 10:A student should answer 3 out of 5 essay questions on a test. In how many different ways can the student select the questions?

Applying Counting Rules to Probability

Ex: The quality assurance engineer of a television manufacturer inspects TVs in lots of 100. He selects 5 of the 100 TVs at random and inspects them thoroughly. Assuming that 6 of the 100 TVs in the current lot are defective, find the probability that exactly 2 of the 5 TVs selected by the engineer are defective.

Homework

p.179 # 4.112 , 4.113p.186 # 4.128 , 4.129, 4.136p.194 # 4.156 , 4,159p.179 # 4.112 , 4.113p.186 # 4.128 , 4.129, 4.136