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Counting Principles and Permutations in MAT 312

Learn about counting principles, such as the Pigeonhole Principle, Addition Principle, Multiplication Principle, and Combinations, and how to apply them in different scenarios in MAT 312.

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Counting Principles and Permutations in MAT 312

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  1. . . . PENNIES for theAGES. . . • Push the “Sample More Data” button on the screen and read the average age of a sample of 36 pennies taken from the jar. • Note the horizontal and vertical scales on the grid here and then record that (rounded) average age using a properly scaled X. MAT 312

  2. Prob & Stat (MAT 312)Dr. Day Tuesday April 1, 2014 • Grab 36 Pennies at Random then Calculateand Plot Average Age • Test #2 and Results • Incoming Task: Assignment #6: Counting • Basic Concepts and Principles of Counting • The Pigeonhole Principle • The Addition Principle • The Multiplication Principle • Permutations • Combinations MAT 312

  3. The Pigeonhole Principle How many students would be required to place soda orders, one soda per student, in order to insure that at least one of the six listed sodas would be ordered by at least two students? • "worst-case scenario" strategy • two variables in the original problem situation • the number of sodas available, and • the number of repeat orders desired. • Using n (here, n = 6) to represent the first value and k (here, k = 2) to represent the second, write a statement that generalizes our solution. We have 10 boxes labeled 1 through 10 into which we place jelly beans. How many jelly beans are required to insure that at least one box contains at least as many jelly beans as the label on the box? MAT 312

  4. The Addition Principle If I order one vegetable from Blaise's Bistro, how many vegetable choices does Blaise offer? • If a choice from Group I can be made in n ways and a choice from Group II can be made in m ways, then the number of choices possible from Group I or Group II is n + m. • Condition:None of the elements in Group I are the same as elements in Group II. • Generalization • Disjoint: have nothing in common. Set I: {a,m,r} Set II: {b,d,i,l,u} Set III: {c,e,n,t} How many ways are there to choose one letter from among the sets I, II, or III? Set A={2,3,5,7,11,13} Set B={2,4,6,8,10,12}. How many ways are there to choose one integer from among the sets A or B? MAT 312

  5. The Multiplication Principle A "meal" at the Bistro consists of one soup item, one meat item, one green vegetable, and one dessert item from the a-la-karte menu. If Blaise's friend Pierre always orders such a meal, how many different meals can be created? • Enumerate the possible meals • How else could we complete the count without identifying all possible options? • If a task involves two steps and the first step can be completed in n ways and the second step in m ways, then there are nm ways to complete the task. • Condition: The ways each step can be completed are independent of each other. • Generalization Set I: {a,m,r} Set II: {b,d,i,l,u} Set III: {c,e,n,t} Determine the number of three-letter sets that can be created such that one letter is from set I, one letter in from set II, and one letter is from set III. MAT 312

  6. Permutations Set I: {a,m,r} Set II: {b,d,i,l,u} Set III: {c,e,n,t} In how many ways can the letters within just one set, from among I, II, and III, be ordered? • Multiplication Principle • Factorial Notation Permutations: ordered arrangements of items. Almost every morning or evening on the news I hear about the State of Illinois DCFS, the Department of Children and Family Services. I get confused, because our mathematics department has a committee called the Department Faculty Status Committee, or DFSC. Can you see why I'm confused? How many different 4-letter ordered arrangements exist for the set of letters {D, F, S, C}? MAT 312

  7. Permutations Extension: Consider ordered arrangements of only some of the elements in a set. If Blaise will post only four possible soda choices, how many different ordered arrangements of the four sodas are there? • Notation: P(n,r): the number of ways to arrange r objects from a set of n objects. • P(4,4) MAT 312

  8. Combinations What is the distinction between asking these two questions? In how many ways can a 5-card poker hand be dealt? How many different 5-card poker hands exist? Permutations: P(52,5) Combinations: Selection of objects from a set with no regard for order or arrangement. Simpler Problem: How many ways can we selectthree items from the 5-element set {A,B,C,D,E} if the order of the three items is disregarded? List the unique 3-element subsets of {A,B,C,D,E}: ABC, ABD, ABE, ACD, ACE, ADE, BCD, BCE, BDE, CDE. There are 10 such 3-element subsets. Or, relate to permutations There are P(5,3) = 60 ordered arrangements of the 5-element set into 3-element subsets. Within the 60 ordered arrangements, there are 10 groups, each with 6 arrangements that use the same 3-letter subset. That is, 60 ÷ 6 = 10 unique 3-element subsets. MAT 312

  9. Combinations Notation Generalization: The number of combinations of n items selected r at a time, where the order of selection or the arrangement of the r items is notconsidered. Note: MAT 312

  10. Assignment #7 • Textbook: Chapter 6, pp 180-181 • #3 • #4 • #5ab • #6a • #7ac • #8ab • #10abc Provide explanation and numerical response! MAT 312

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