Objectives Solve counting problems using the Multiplication Rule Solve counting problems using permutations Solve counting problems using combinations

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  • ObjectivesSolve counting problems using the Multiplication Rule

    Solve counting problems using permutations

    Solve counting problems using combinations

    Solve counting problems involving permutations with nondistinct items

    Compute probabilities involving permutations and combinations

  • Do Now:Write down all the items you would put in your fruit salad for the end of the year picnic :)- New VOCABulary CombinationPermutationFactorialPractice ProblemsBlock 2Midterm discussion & UpdateGo over Tuesdays HW

    Objectives:

    Solve problems using permutations

    Solve problems using combinations

  • Precalculus 2!!!

    over your midterm are there questions you have or points you believe you deservespecifically, look at your answers for permutations and combinations

    DO NOW: Notes on the comparisons

    Portfolio discussion

    QUIZ next week!

    Objectives:Define permutation, combination, factorial, favorable outcome

    Solve probability, permutation and combination problemsHW: Portfolio!

  • Do Now:Write down all the items you would put in your fruit salad for the end of the year picnic :)

    order does not matter

    How about your locker or bike lock combination?Does order matter??!! Objectives:

    Solve problems using permutations

    Solve problems using combinations

  • Class electionOk, so we have 16 students in the class and we are going to make a council of 3 student representatives. A pres, a vice pres and a treasure.

    How many ways can we create our trio?

    Does order matter?

  • Class electionOk, so we have 16 students in the class and we are going to make a council of 3 student representatives of equal value?

    How many ways can we create our trio?

    Does order matter?

  • Vocabulary

    Permutation is an ordered arrangement in which r objects are chosen from n distinct (different) objects and repetition is not allowed. The symbol nPr represents the number of permutations of r objects selected from n objects.

    Combination is a collection, without regard to order, of n distinct objects without repetition. The symbol nCr represents the number of combinations of n objects taken r at a time..

  • Vocabulary

    Permutation is an ordered arrangement in which r objects are chosen from n distinct (different) objects and repetition is not allowed. The symbol nPr represents the number of permutations of r objects selected from n objects.

    How would I represent the class election example?!

    Combination is a collection, without regard to order, of n distinct objects without repetition. The symbol nCr represents the number of combinations of n objects taken r at a time.

    How would I represent the class election example?!

  • Vocabulary Fundamental Counting Principle Use this when want to find the total ways (number) a task can occur. (license plate problem) Multiply! (ABCDEF) = n(A) * n(B) *n(C) * n(D) * n(E) * n(F)

  • Vocabulary Handout Ok, so in your own words definePermutation and Combination

    Draw an example to help you remember

    -you can wait on rating my understanding until the end of class

  • PermutationsNumber of Permutations of n Distinct Objects taken r at a time:N objects are distinctOnce used an object cannot be repeatedOrder is important n!nPr = ----------- (n r)!

    Factorial n! is defined to be n! = n(n-1)(n-2)(n-3).. (3)(2)(1)

  • PermutationsNumber of Permutations of n Distinct Objects taken r at a time:N objects are distinctOnce used an object cannot be repeatedOrder is important n!nPr = ----------- (n r)!

    How would I represent the class election example?!

  • CombinationsNumber of Combinations of n Distinct Objects taken r at a time:N objects are distinctOnce used an object cannot be repeated (no repetition)Order is not important

    n!nCr = ----------- r!(n r)!

    How would I represent the class election example?!

  • CombinationsNumber of Combinations of n Distinct Objects taken r at a time:N objects are distinctOnce used an object cannot be repeated (no repetition)Order is not important

    n!nCr = ----------- r!(n r)!

  • CombinationsNumber of Combinations of n Distinct Objects taken r at a time:N objects are distinctOnce used an object cannot be repeated (no repetition)Order is not important

    n!nCr = ----------- r!(n r)!

    r! in the denominator eliminates the double count!!In other words, ABC and BAC are the SAME in a combination! *in a permutation, they are differentthink of pres, vp and treas

  • How to TellIs a problem a permutation or a combination?

    One way to tellWrite down one possible solution (i.e. Roger, Rick, Randy)Switch the order of two of the elements (i.e. Rick, Roger, Randy)

    Is this the same result?If no this is a permutation order mattersIf yes this is a combination order does not matter

  • Vocabulary Handout Ok, so now how about your own exampleyou may put this is the further understanding box

    And rate my understanding 1 is the least (amount of understanding )and 4 is the most

  • Your TurnPartner Up pairs of 2 or groups of 3 and work on CWI will be coming around to help and check

    We MISSED TWO DAYS AND I WAS IN A MEETING TUESDAY SO THIS IS OUR ONLY DAY THIS WEEK AND WE NEED TO STAY FOCUSEDgrrrrreat

  • Example 1If there are 3 different colors of paint (red, blue, green) that can be used to paint 2 different types of toy cars (race car, police car), then how many different toys can there be?

  • Example 1 IllustratedA tree diagram of the different possibilities

    PaintCarPossibilities

  • Example 2In a horse racing Trifecta, a gambler must pick which horse comes in first, which second, and which third. If there are 8 horses in the race, and every order of finish is equally likely, what is the chance that any ticket is a winning ticket?

    The probability that any one ticket is a winning ticket is 1 out of 8P3, or 1 out of 336

  • Permutations with replacementNumber of Permutations of n Distinct Items taken r at a time with replacement:N objects are distinctOnce used an object can be repeated (replacement)Order is important P = nr

  • Example 3Suppose a computer requires 8 characters for a password. The first character must be a letter, but the remaining seven characters can be either a letter or a digit (0 thru 9). The password is not case-sensitive. How many passwords are possible on this computer?26 367 = 2.037 x 1012

  • Example 4If there are 8 researchers and 3 of them are to be chosen to go to a meeting, how many different groupings can be chosen?

  • Permutations non-distinct itemsNumber of Permutations with Non-distinct Items:N objects are not distinctK different groups

    n!P = --------------------- where n = n1 + n2 + + nk n1!n2! .nk!

  • Example 5How many different vertical arrangements are there of 9 flags if 4 are white, 3 are blue and 2 are red? 9! 987654! 98765----------- = ------------------ = --------------- = 1260 4!3!2! 4!3!2! 32121

  • Permutation vs CombinationComparing the description of a permutation with the description of a combination

    The only difference is whether order matters

    PermutationCombinationOrder mattersOrder does not matterChoose r objectsChoose r objectsOut of n objectsOut of n objectsNo repetitionNo repetition

  • Multiplication Rule of CountingIf a task consists of a sequence of choices in which there are p selections for the first item, q selections for the second item, and r choices for the third item, and so on, then the task of making these selections can be done in p q r .. different ways

    The classical method, when all outcomes are equally likely, involves counting the number of ways something can occurThis section includes techniques for counting the number of results in a series of choices, under several different scenarios

  • VocabularyFactorial n! is defined to be n! = n(n-1)(n-2)(n-3).. (3)(2)(1)

    Permutation is an ordered arrangement in which r objects are chosen from n distinct (different) objects and repetition is not allowed. The symbol nPr represents the number of permutations of r objects selected from n objects.

    Combination is a collection, without regard to order, of n distinct objects without repetition. The symbol nCr represents the number of combinations of n objects taken r at a time..

  • Summary and HomeworkSummaryThe Multiplication Rule counts the number of possible sequences of itemsPermutations and combinations count the number of ways of arranging items, with permutations when the order matters and combinations when the order does not matterPermutations and combinations are used to compute probabilities in the classical method

    Homework

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