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Counting

Counting. 11.3. Apply the fundamental counting principle Calculate and apply permutations Calculate and apply combinations. Tree Diagram.

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Counting

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  1. Counting 11.3 Apply the fundamental counting principle Calculate and apply permutations Calculate and apply combinations

  2. Tree Diagram Suppose that a quiz has only two questions. The first is a multiple-choice question withfour choices, A, B, C, or D, and the second is a true-false (T-F) question. A tree diagram can be used to count the ways that this quiz can be answered.

  3. Tree Diagram A tree diagram is a systematic way of listing every possibility. Wecan see that there are eight ways to answer the test. They areAT, AF, BT, BF, CT, CF, DT, and DF.For instance, CF indicates a quiz with answers of C on the first question and F on the second question.

  4. Fundamental Counting Principle Let E1, E2, E3,…, Enbe a sequence of n independent events. If event Ek can occur mk ways for k=1, 2, 3,…, n,then there are m1•m2•m3•… •mn ways for all n events to occur. Two events are independent if neither event influences the outcome of the other.

  5. An exam contains four true-false questions and six multiple-choice questions. Each multiple-choice question has five possible answers. Count the number of ways that the exam can be answered. Solution This is a sequence of ten independent events. There are two ways to answer each of the first four questions. There are five ways to answer the next six questions. Example: Counting ways to answer an exam

  6. The number of ways to answer the exam is Example: Counting ways to answer an exam

  7. Sometimes a license plate is limited to 3 uppercase letters (A through Z) followed by 3 digits (0 through 9). For example, ABB 112 would be a valid license plate. Would this formatprovide enough license plates for a state with 8 million vehicles? Example: Counting license plates

  8. 26 choices for letters 10 choices for digits There are 26 ways to choose each of the three letters and 10 ways to choose each of the 3 digits. This format for license plates could accommodate more than 8 million vehicles. Example: Counting license plates

  9. Count the total number of 800 numbers if the local portion of a telephone number (the last seven digits) does not start with a 0 or 1. Solution A toll-free 800 number assumes the following form. Example: Counting toll-free telephone numbers

  10. We can think of choosing the remaining digits for the local number as seven independent events. Since the local number cannot begin with a 0 or 1, there are eight possibilities(2 to 9) for the first digit. The remaining six digits can be any number from 0 to 9, sothere are ten possibilities for each of these digits. The total is given by Example: Counting toll-free telephone numbers

  11. n-Factorial For any natural number n, n! = n(n – 1)(n – 2) … (3)(2)(1) and 0! = 1.

  12. Compute n! for and n = 0, 1, 2, 3, 4,5 by hand.Use a calculator to find 8!, 13!, and 25!. Solution 0! = 11! = 1 2! = 2 1 = 2 3! = 3  2 1 = 6 4! = 4 3  2 1 = 24 5! = 5 4 3  2 1 = 120 Example: Calculating factorials

  13. Here’s the calculator display. Note that the value for 25! is an approximation. Notice how rapidly n! increases! Example: Calculating factorials

  14. Permutation A permutation is an ordering or arrangement.

  15. Permutations of n ElementsTaken r at a Time If P(n, r)denotes the number of permutations of n elements taken r at a time, withr≤ n,then

  16. For a class of 30 students, how many arrangements are there in which 4 students each givea speech? Solution The number of permutations of 30 elements taken 4 at a time is given by There are 657,720 ways to arrange the four speeches. Example: Calculating permutations

  17. Combination A combination is not an ordering or arrangement, but rather a subset of a set of elements. Order is not important when finding combinations. The number of possible subsets, or combinations, is denoted either C(4, 2) or

  18. Combinations of n ElementsTaken r at a Time If C(n, r)denotes the number of combinations of n elements taken r at a time, withr≤ n,then

  19. Calculate each of the following. Support your answer by using a calculator. Solution Example: Calculating C(n,r)

  20. Example: Calculating C(n,r)

  21. A college student has five courses left in her major and plans to take two of them this semester.Assuming that this student has the prerequisites for all five courses, determine how manyways these two courses can be selected. Example: Counting combinations

  22. Solution The order in which the courses are selected is unimportant. From a set of 5courses, the student selects a subset of 2 courses. The number of subsets is There are 10 ways to select two courses from a set of five. Example: Counting combinations

  23. How many committees of six people can be selected from six women and three men, if a committee must consist of at least two men? Solution Two Men: Committee would be two men and four women. Example: Counting committees

  24. Three Men: Committee would be three men and three women The total number of possible committees would be45 + 20 = 65. Example: Counting committees

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