Ch Counting Principles

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

Example 2  Eight pieces of paper are numbered 1-8 and placed in a box. Two pieces of paper are drawn at random from the box at the same time, and the numbers on the pieces of paper are written down and totaled. How many different ways can a sum of 12 be obtained.

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

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

Example 3  How many different pairs of letters from the English alphabet are possible?

Example 4  Telephone numbers in the United States currently have 10 digits. The first three are the area code and the next seven are the local telephone number. How many different telephone numbers are possible within each are code? (Note that at this time, a local telephone number cannot begin with 0 or 1)

Permutations  Used to determine “the number of ways that n elements can be arranged”  Def: permutation – a permutation on n elements is an ordering of the elements such that one element is first, on is second, on is third and so on

Example 5  How many permutations are possible for the letters A, B, C, D, E, and F??

Example 6  Eight horses are running in a race. In how many different ways can these horses come in first, second or third?

Permutations of n Elements Taken R at a time  The number of permutations of n elements taken r at a time is P(n,r) = __n!__ (n-r)! (n-r)!

Distinguishable Permutations  Suppose a set of n objects has n1 of one kind of object, n2 of a second kind and so on with n = n1+n2+n3+n4 +….nk Then the number of distinguishable permutations of the n objects is ______n!_____n1!n2!n3!....nk!

Example 7  In how many distinguishable ways can the letters in BANANA be written?

Combinations  Selecting items of a larger set in which order is NOT important The number of combinations of n elements taken r at a time is C(n,r) = __n!__ (n-r)! r!

Example 8  In how many different ways can three letters be chosen from the letters A, B, C, D, and E?

Example 9  A standard poker hand consists of five cards dealt from a deck of 52 cards. How many different poker hands are possible? (After the cards are dealt, the player may reorder them, so order is not important)

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

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