Objectives: Permutations. Fundamental Counting Principle. Fundamental Counting Principle. Fundamental Counting Principle
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1 and Objectives:! apply fundamental counting principle! compute permutations! compute combinations HL2 Math - Santowski! distinguish permutations vs combinations can be used determine the number of possible outcomes when there are two or more characteristics. states that if an event has m possible outcomes and another independent event has n possible outcomes, then there are m * n possible outcomes for the two events together. Lets start with a simple example. A student is to roll a die and flip a coin. How many possible outcomes will there be? Lets start with a simple example. A student is to roll a die and flip a coin. How many possible outcomes will there be? 1H 2H 3H 4H 5H 6H 1T 2T 3T 4T 5T 6T 6*2 = 12 outcomes For a college interview, Robert has to choose what to wear from the following: 4 slacks, 3 shirts, 2 shoes and 5 ties. How many possible outfits does he have to choose from? 12 outcomes 1
2 For a college interview, Robert has to choose what to wear from the following: 4 slacks, 3 shirts, 2 shoes and 5 ties. How many possible outfits does he have to choose from? 4*3*2*5 = 120 outfits A Permutation is an arrangement of items in a particular order. Notice, ORDER MATTERS! To find the number of of n items, we can use the Fundamental Counting or factorial notation. The number of ways to arrange the letters ABC: Number of choices for first blank? 3 Number of choices for second blank? 3 2 Number of choices for third blank? *2*1 = 6 3! = 3*2*1 = 6 ABC ACB BAC BCA CAB CBA To find the number of of 5 p 3 To find the number of of A combination lock will open when the right choice of three numbers (from 1 to 30, inclusive) is selected. How many different lock combinations are possible assuming no number is repeated? 2
3 A combination lock will open when the right choice of three numbers (from 1 to 30, inclusive) is selected. How many different lock combinations are possible assuming no number is repeated? From a club of 24 members, a President, Vice President, Secretary, Treasurer and Historian are to be elected. In how many ways can the offices be filled? From a club of 24 members, a President, Vice President, Secretary, Treasurer and Historian are to be elected. In how many ways can the offices be filled? A Combination is an arrangement of items in which order does not matter. ORDER DOES NOT MATTER! Since the order does not matter in combinations, there are fewer combinations than permutations. The combinations are a "subset" of the permutations. To find the number of of To find the number of of 5C 3 3
4 To find the number of of To play a particular card game, each player is dealt five cards from a standard deck of 52 cards. How many different hands are possible? To play a particular card game, each player is dealt five cards from a standard deck of 52 cards. How many different hands are possible? A student must answer 3 out of 5 essay questions on a test. In how many different ways can the student select the questions? A student must answer 3 out of 5 essay questions on a test. In how many different ways can the student select the questions? A basketball team consists of two centers, five forwards, and four guards. In how many ways can the coach select a starting line up of one center, two forwards, and two guards? 4
5 A basketball team consists of two centers, five forwards, and four guards. In how many ways can the coach select a starting line up of one center, two forwards, and two guards? Center: Forwards: Guards: Guidelines on Which Method to Use Thus, the number of ways to select the starting line up is 2*10*6 =
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