Ch. 11.6 Ch. 11.6

Counting PrinciplesCounting Principles

Example 1Example 1

Eight pieces of paper are numbered from Eight pieces of paper are numbered from 1-8 and placed in a box. One piece of 1-8 and placed in a box. One piece of paper is drawn from the box, its number is paper is drawn from the box, its number is written down, and the piece of paper is written down, and the piece of paper is replaced in the box.replaced in the box. Then, a second piece Then, a second piece of paper is drawn from the box and its of paper is drawn from the box and its number is written down. Finally, the two number is written down. Finally, the two numbers are added together. How many numbers are added together. How many different ways can a sum of twelve be different ways can a sum of twelve be obtained????obtained????

Example 2Example 2

Eight pieces of paper are numbered 1-8 Eight pieces of paper are numbered 1-8 and placed in a box. Two pieces of paper and placed in a box. Two pieces of paper are drawn at random from the box are drawn at random from the box at the at the same timesame time, and the numbers on the , and the numbers on the pieces of paper are written down and pieces of paper are written down and totaled. How many different ways can a totaled. How many different ways can a sum of 12 be obtained.sum of 12 be obtained.

Ex 1 and 2Ex 1 and 2

The difference between examples 1 and The difference between examples 1 and 2 is that example 1 occurred WITH 2 is that example 1 occurred WITH REPLACEMENT and example 2 REPLACEMENT and example 2 occurred WITHOUT REPLACEMENT occurred WITHOUT REPLACEMENT (this eliminates extra possibilities)(this eliminates extra possibilities)

Fundamental Counting Fundamental Counting PrinciplePrinciple

Let E1 and E2 be events. The first event Let E1 and E2 be events. The first event E1 can occur in m1 different ways. After E1 can occur in m1 different ways. After E1 has occurred, E2 can occur in m2 E1 has occurred, E2 can occur in m2 different ways. The number of ways that different ways. The number of ways that the two events can occur is m1 m2the two events can occur is m1 m2

Example 3Example 3

How many different pairs of letters from How many different pairs of letters from the English alphabet are possible?the English alphabet are possible?

Example 4Example 4

Telephone numbers in the United States Telephone numbers in the United States currently have 10 digits. The first three currently have 10 digits. The first three are the area code and the next seven are are the area code and the next seven are the local telephone number. How many the local telephone number. How many different telephone numbers are possible different telephone numbers are possible within each are code? (Note that at this within each are code? (Note that at this time, a local telephone number cannot time, a local telephone number cannot begin with 0 or 1)begin with 0 or 1)

PermutationsPermutations

Used to determine “the number of ways Used to determine “the number of ways that n elements can be arranged”that n elements can be arranged”

Def: permutation – a permutation on n Def: permutation – a permutation on n elements is an ordering of the elements elements is an ordering of the elements such that one element is first, on is such that one element is first, on is second, on is third and so onsecond, on is third and so on

Example 5Example 5

How many permutations are possible for How many permutations are possible for the letters A, B, C, D, E, and F??the letters A, B, C, D, E, and F??

Example 6Example 6

Eight horses are running in a race. In Eight horses are running in a race. In how many different ways can these how many different ways can these horses come in first, second or third?horses come in first, second or third?

Permutations of n Elements Permutations of n Elements Taken R at a timeTaken R at a time

The number of permutations of n The number of permutations of n elements taken r at a time is elements taken r at a time is

P(n,r) = P(n,r) = __n!____n!__

(n-r)!(n-r)!

Distinguishable Distinguishable PermutationsPermutations

Suppose a set of n objects has n1 of one Suppose a set of n objects has n1 of one kind of object, n2 of a second kind and so kind of object, n2 of a second kind and so on with on with

n = n1+n2+n3+n4 +….nkn = n1+n2+n3+n4 +….nk

Then the number of distinguishable Then the number of distinguishable permutations of the n objects is permutations of the n objects is

______n!___________n!_____

n1!n2!n3!....nk!n1!n2!n3!....nk!

Example 7Example 7

In how many distinguishable ways can In how many distinguishable ways can the letters in BANANA be written?the letters in BANANA be written?

CombinationsCombinations

Selecting items of a larger set in which Selecting items of a larger set in which order is NOT importantorder is NOT important

The number of combinations of n The number of combinations of n elements taken r at a time is elements taken r at a time is

C(n,r) = C(n,r) = __n!____n!__

(n-r)! r!(n-r)! r!

Example 8Example 8

In how many different ways can three In how many different ways can three letters be chosen from the letters A, B, C, letters be chosen from the letters A, B, C, D, and E?D, and E?

Example 9Example 9

A standard poker hand consists of five A standard poker hand consists of five cards dealt from a deck of 52 cards. cards dealt from a deck of 52 cards. How many different poker hands are How many different poker hands are possible?possible?

(After the cards are dealt, the player may (After the cards are dealt, the player may reorder them, so order is not important)reorder them, so order is not important)

Example 10Example 10

You are forming a 12 member swim team You are forming a 12 member swim team from 10 girls and 15 boys. The team from 10 girls and 15 boys. The team must consist of five girls and seven boys. must consist of five girls and seven boys. How many different 12 member teams How many different 12 member teams are possible?are possible?

HomeworkHomework

Pg 860 #7-17, 21,37-39, 41,42, 49-55Pg 860 #7-17, 21,37-39, 41,42, 49-55