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Fundamental Counting PrincipleLets 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 6*2 = 12 outcomes1T 2T 3T 4T 5T 6T 12 outcomes
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Fundamental Counting PrincipleFor a college interview, Robert has to choosewhat to wear from the following: 4 slacks, 3shirts, 2 shoes and 5 ties. How many possibleoutfits does he have to choose from? 4*3*2*5 = 120 outfits
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Permutations A Permutation is an arrangement of items in a particular order. Notice, ORDER MATTERS!To find the number of Permutations ofn items, we can use the FundamentalCounting Principle or factorial notation.
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Permutations 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? 3 2 1 3*2*1 = 6 3! = 3*2*1 = 6ABC ACB BAC BCA CAB CBA
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PermutationsTo find the number of Permutations ofn items chosen r at a time, you can usethe formula n! n pr = ( n âˆ’ r )! where 0 â‰¤ r â‰¤ n . 5! 5!5 p3 = = = 5 * 4 * 3 = 60 (5 âˆ’ 3)! 2!
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PermutationsPractice: 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? Answer Now
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PermutationsPractice: 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? 30! 30!30 p3 = = = 30 * 29 * 28 = 24360 ( 30 âˆ’ 3)! 27!
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PermutationsPractice: 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? Answer Now
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PermutationsPractice: 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? 24! 24! 24 p5 = = = ( 24 âˆ’ 5)! 19! 24 * 23 * 22 * 21 * 20 = 5,100,480
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Combinations A Combination is an arrangement of items in which order does not matter.ORDER DOES NOT MATTER!Since the order does not matter incombinations, there are fewer combinationsthan permutations. Â The combinations are a"subset" of the permutations.
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Combinations To find the number of Combinations of n items chosen r at a time, you can use the formula n! C = where 0 â‰¤ r â‰¤ n .n r r! ( n âˆ’ r )!
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CombinationsTo find the number of Combinations ofn items chosen r at a time, you can usethe formula n! C = where 0 â‰¤ r â‰¤ n . n r r! ( n âˆ’ r )! 5! 5! 5 C3 = = = 3! (5 âˆ’ 3)! 3!2! 5 * 4 * 3 * 2 * 1 5 * 4 20 = = = 10 3 * 2 *1* 2 *1 2 *1 2
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CombinationsPractice: 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? Answer Now
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CombinationsPractice: 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? 52! 52! 52 C5 = = = 5! (52 âˆ’ 5)! 5!47! 52 * 51 * 50 * 49 * 48 = 2,598,960 5* 4* 3* 2*1
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CombinationsPractice: A student must answer 3 out of 5 essay questions on a test. In how many different ways can the student select the questions? Answer Now
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CombinationsPractice: A student must answer 3 out of 5 essay questions on a test. In how many different ways can the student select the questions? 5! 5! 5 * 4 5 C3 = = = = 10 3! (5 âˆ’ 3)! 3!2! 2 * 1
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CombinationsPractice: 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? Answer Now
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CombinationsPractice: 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: 2! 5! 5 * 4 4! 4 * 3 2 C1 = =2 5 C2 = = = 10 4 C2 = = =6 1!1! 2!3! 2 * 1 2!2! 2 * 1 2 C1 * 5 C 2 * 4 C 2 Thus, the number of ways to select the starting line up is 2*10*6 = 120.
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