The order in which the officers are chosen matters. There is one set of questions.P_3\) on the TI calculator, type the value of \(n\), go to the MATH menu and move right to the PRB sub-menu, select the \(_nP_r\) command, type the value of \(r\), and press ENTER. Indicate which of the following statements are correct and which are not.Ĭlick Yes or No, then Submit. Applying the Multiplication Principle one can see that there are 2 n subsets of A. Thus there are n events and each event has two possible outcomes. In this case selecting or not selecting an element is an event. That is, for a subset, say B, of A, each element of A is either selected or not selected into B. A subset of A can be constructed by selecting elements of A. Thus altogether 26 * 25* 24* 23 = 358,800 possible labels exist.Įxample 3: The number of subsets of a finite set can be computed using the Multiplication Principle. Similarly the third letter is now selected from the remaining twenty four letters and the fourth from twenty three letters. The second letter, however, must be selected from twenty five letters because one letter has been selected for the first position and that letter can not be used for the second position. Then the first letter of a label can be selected from all twenty six letters. Suppose that no letters can be duplicated in a label. Then there are 3 * 5 = 15 major routes from Washington DC to Los Angeles.Įxample 2: Suppose that four upper case letters are used for labels. * n k.Įxample 1: Suppose that there are three major auto routes from Washington DC to Chicago, and five from Chicago to Los Angeles. Counting Principles Learning Outcomes Solve counting problems using the Addition Principle and the Multiplication Principle. Then the total number of outcomes for the sequence of these k events is n 1 * n 2 *. Then the total number of outcomes for the sequence of the two events is n 1 * n 2. Let A 1 and A 2 be events with n 1 and n 2 possible outcomes, respectively. In general the Multiplication Principle of Counting is stated as follows: As in the case of the Addition Principle, selecting one of five (four or three) choices is called an event, and a specific clock rate (RAM size or disk capacity) is called the outcome of the event. This is the Multiplication Principle of Counting. In high school, permutations and combinations are emphasized in Integrated Math II (or Algebra II) and the Math Analysis (precalculus) courses. Thus there are 5 * 4 * 3 = 60 different combinations of configurations altogether. The Fundamental Counting Principle is introduced in elementary and middle school and forms the foundation for enumerating quantities given varying choices. Then for each of those 20 combinations there are three different disk capacities. Then for each selection of clock rate one can choose one of four RAM sizes giving 5 * 4 different combinations of clock rate and RAM size. The question is how many different choices of computer there are assuming that everything else is the same for simplicity.įirst, there are five possible clock rates. + n k.Ĭonsider the following problem of purchasing a computer: A manufacturer A offers five different CPU clock rates, four different sizes of RAMs and three different capacities of disks among others. Then the total number of outcomes for the event "A 1 or A 2 or. This addition principle can be generalized for more than two events. According to the addition principle there are 5 + 8 = 13 possible selections. There are 5 oucomes for the chicken event and 8 outcomes for the beef event. In this case, an event is "selecting a dish of either kind". How many selections does a customer have ? Note that the events must be disjoint, that is they must not have common outcomes for this principle to be applicable.Įxample: Suppose there are 5 chicken dishes and 8 beef dishes. Then the total number of outcomes for the event "A 1 or A 2" is n 1 + n 2. Let A 1 and A 2 be disjoint events, that is events having no common outcomes, with n 1 and n 2 possible outcomes, respectively. Thus the event "selecting one from make A 1", for example, has 12 outcomes. Choosing one from given models of either make is called an event and the choices for either event are called the outcomes of the event. This is the Addition Principle of Counting. Since we can choose one of 12 models of make A 1 or one of 18 of A 2, there are altogether 12 + 18 = 30 models to choose from. Then how many models are there altogether to choose from ? Suppose that we want to buy a computer from one of two makes A 1 and A 2 Suppose also that those makes have 12 and 18 different models, respectively.
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