Quiz Plot the point: (-4, 2, -3) in the Cartesian space. Find the midpoint between the 2 points: P(1, 5, -7) and Q(-5, 3, -3) 3. Find the distance between (-1, 2, 5) and (3, -4, 6) (-1, 2, 5) and (3, -4, 6) 4. What is the radius and center point of the sphere? Q (-3, -3, 4 ) P (2, 3, 4) 5. What is the vector from “P” to “Q” ?

Chapter 9 Discrete Mathematics

HOMEWORK Section 9-1 (page 708) (evens) 2-34, 38

9-1 Basic Combinatorics This section does not specifically prepare you for Calculus, BUT, when/where else are you you for Calculus, BUT, when/where else are you going to learn this? going to learn this?

What you’ll learn about Discrete Versus Continuous The Multiplication Principle of Counting Permutations Combinations … and why Counting sets is important in insurance, gambling, when you are trying to find out if you even have a chance of winning ($, trophy..)

Vocabulary: Continuous Data: Data that contains an infinite number of points within a data range. number of points within a data range. Example: number line, there are an infinite number of numbers in the range [3, 5]. of numbers in the range [3, 5]. Continuous data does not lend itself very well to “counting.” to “counting.”

Vocabulary: Discrete Data: Data sets that contain a finite number of data points. number of data points. Example: The number of ways you can assign students to seats in this class. students to seats in this class. Discrete mathematics deals with counting.

Given the first letter above, the second letter could be: Arranging 3 Objects in Order How many way can you arrange the letters A, B, and C ? A, B, and C ? A B C B or C A or C B or A Any one of the following 3 could be the 1 st letter. The only option for the 3 rd letter in each case is: C B C A A B

Arranging 3 Objects in Order How many way can you arrange the letters A, B, and C ? A, B, and C ? A B C B C A C B A C B C A A B ABC,ACB,BAC,BCA,CBA,CAB SIX ways We call this a tree diagram.

Your Turn: 1.How many ways could you arrange 4 people in a line? in a line? This starts getting too laborious when you have more and more people to line up. (How many more and more people to line up. (How many ways could you arrange all the people in this ways could you arrange all the people in this class in a line?) class in a line?) Luckily, we have the multiplication principle.

Multiplication Principle of Counting (aka) The Fundamental Counting Principle

Using the Fundamental counting Principle to count the # of ways to line up 3 people. The first position in line could be either 1 of 3 people 3 ways. The second position in line could be either 1 of 2 people 2 ways. The last position in line could only be the person left over. Total number of ways = 3 * 2 * 1 = 3! “!” means “factorial” Since you “use a person up” (no replacement) each subsequent position has 1 less possibility than the subsequent position has 1 less possibility than the previous position. previous position. 1 way.

The Counting Principle: Arranging People: You have 1 less person to use in each subsequent use in each subsequent step. step. We call this “arranging without replacement” Arranging Numbers and Letters on a license plate: the previous number or letter can be used again: We call this “arranging with replacement”.

With Replacement or Without Replacement Your turn: which is it for: 2.Assigning 3 committee members to the positions of: “Pres”, “Vice-Pres”, and “Secretary” “Pres”, “Vice-Pres”, and “Secretary” 3. Assigning Social security numbers to an individual

Using the Multiplication Principle If a license plate has four letters followed by three numerical digits. Find the number of different license plates that could be formed if there is no restriction on the letters or digits that can be used. # # # A B C How many possibilities for the first number? 10

Using the Multiplication Principle If a license plate has four letters followed by three numerical digits. Find the number of different license plates that could be formed if there is no restriction on the letters or digits that can be used. # # # A B C How many possibilities for the 2nd number? 10 * 10

Using the Multiplication Principle If a license plate has four letters followed by three numerical digits. Find the number of different license plates that could be formed if there is no restriction on the letters or digits that can be used. # # # A B C How many possibilities for the 3rd number? 10 * 10

Using the Multiplication Principle If a license plate has four letters followed by three numerical digits. Find the number of different license plates that could be formed if there is no restriction on the letters or digits that can be used. # # # A B C How many possibilities for the 1 st letter? 10 * 10 * 26

Using the Multiplication Principle If a license plate has four letters followed by three numerical digits. Find the number of different license plates that could be formed if there is no restriction on the letters or digits that can be used. # # # A B C How many possibilities for the 2nd letter? 10 * 10 * 26

Using the Multiplication Principle If a license plate has four letters followed by three numerical digits. Find the number of different license plates that could be formed if there is no restriction on the letters or digits that can be used. # # # A B C How many possibilities for the 3rd letter? 10 * 10 * 26 = 17,576,000 Wow!

Permutations One important application of the Fundamental Counting Principle is in determining the number of ways that n element can be arranged (in order). An ordering of n elements is called a permutation of the elements. A permutation of n different elements is an ordering of the elements such that one element is first, one is second, one is third, and so on. (Replacement or no replacement ?)

Permutations of an n-Set How many permutations are possible for the letters A, B, C, D, E, and F? The number of permutations of n elements is given by n (n – 1) (n – 2) … = n! In other words, there are n! different ways that n elements can be ordered.

Your Turn: Count the number of different 8-letter “words” (groups of 8 letters) that can be formed using the letters in the word COMPUTER. Each permutation of the 8 letters forms a different word. There are 8! = 40,320 such permutations. 4.

What if two of the letters are the same? Count the number of different 8-letter “words” that can be formed using the letters in the word “WAAG”. Let “A” be the 1 st A. Let “A” be the 2 nd A. What’s the difference between AAWG and AAWG? There’s no difference!! They are not distinguishable from each other. So we really have “double counted” from each other. So we really have “double counted” a bunch of words.

What if two of the letters are the same? Count the number of different 8-letter “words” that can be formed using the letters in the word “WAAG”. AAWG (AAWG) is one example of double counting. To remove the “double counting” we must divide the possible ways to permutate A and A divide the possible ways to permutate A and A AWAG (AWAG) is another example of double counting. We must divide by 2!. When we write an ‘A’ we must choose between 2 A’s, then there is only 1 A left for the second time we use an A.

Distinguishable Permutations 1! = ? For WAAG # of W’s and G’s =1 Distinguishable Permutations = 12

Permutations Counting Formula (We have “n” distinct number of objects to place into “r” number of positions). to place into “r” number of positions). The number of permutations of “n” objects taken “r” at a time, is denoted by: and Oh, by the way: 0! = 1

Arranging 3 Objects in Order How many way can you arrange the letters A, B, and C ? A, B, and C ? ABC,ACB,BAC,BCA,CBA,CAB SIX ways Remember this? 3 items taken 3 at a time. Oh, by the way: 0! = 1

Permutations Counting Formula (We have “n” number of objects to place into “r” number of positions). Your turn: 5.

Permutations Pick 5 cards from a deck. Is there a difference between: Is there a difference between: a) “Ace of hearts” as the first card b) “Ace of hearts as the 4 th card b) “Ace of hearts as the 4 th card If we were making a permutation using the letters ‘D’, ‘A’, ‘W’, and ‘G’ and ‘D’, ‘A’, ‘W’, and ‘G’ and WADGDAWG would be two distinct words. would be two distinct words. ORDER MATTERS!! (with permutations)  a different order of members is a different group all together!! So we must distinguish between groups of things where order matters (will give unique answers where order matters (will give unique answers for the same items but in different order), and when for the same items but in different order), and when order doesn’t matter (5 hearts order doesn’t matter (5 hearts Order doesn’t doesn’t matter!! matter!!

“Order Matters” vs. “Order Doesn’t Matter” Different order of the same items  counted as separate items  counted as separate items Different order of the same items  is not counted as separate items  is not counted as separate items (must divide out the double counting caused by a permutation of items  which counts different order of items as separate/different.

“Order Matters” vs. “Order Doesn’t Matter” Different order  separate items Different Order  not separate items Different Order  not separate items (must divide out the double counting The symbol for this is: “ ‘n’ choose ‘r’ items” “ ‘n’ items taken ‘r’ at a time” “Combination” “Permutation”

Your turn: 6. “ ‘7’ choose ‘3’ items” = ?

Distinguishing between: Permutations and Combinations Permutation: (different order  counted separately) Combination: (different order  not counted separately) 1.Pres, Vice Pres, and Secretary chosen out of 25 candidates. Which is it? Order of being chosen matters  Permutation 2. Farmer picks 5 sheep to be sheered in order to sell their wool. Which is it? Order of being chosen doesn’t matter  Combination.

Permutations or Combinations? person committees formed from a group of 20 people Your turn: (which is it?) 8. 1 st, 2 nd, and 3 rd place trophies awarded to the top three contestants of 100 entrants. 9.The first is a combination (order of being picked doesn’t matter. How many combinations picked doesn’t matter. How many combinations are there? are there? 10. The second is a permutation (order of being picked will give you a different award) How many picked will give you a different award) How many possible groups of winners are there? possible groups of winners are there?