COMBINATIONS & PERMUTATIONS. EATING…something we all love! LUNCH SPECIAL! Choose 1 of each ENTRÉESIDE DISH PizzaColeslaw ChickenSalad BeefFruit DRINK.

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Presentation transcript:

COMBINATIONS & PERMUTATIONS

EATING…something we all love! LUNCH SPECIAL! Choose 1 of each ENTRÉESIDE DISH PizzaColeslaw ChickenSalad BeefFruit DRINK Milk, Punch, Juice, Iced tea

FUNDAMENTAL COUNTING PRINCIPAL If there are n items and m 1 ways to choose the first item, m 2 ways to choose the second item after the first item has been chosen, and so on, then there are m 1 * m 2 * … * m n ways to choose n items. # Options Thing 1 * # Options Thing 2 * # Options Thing 3

IS THERE AN EASIER WAY TO DO THIS? Problem #1 Caillou has to choose his morning outfit. He has 3 pairs of pants (jeans, khakis, and sweat pants) and 4 shirts (green, red, yellow, and blue). HOW MANY OUTFIT OPTIONS DOES CAILLOU HAVE?

Problem #2 – Lunch Again! LUNCH SPECIAL! Choose 1 of each ENTRÉESIDE DISH PizzaColeslaw ChickenSalad BeefFruit DRINK Milk, Punch, Juice, Iced tea

Problem #3: Practicing the Fundamental Counting Principle In Utah, a license plate consists of 3 digits followed by 3 letters. The letters I, O, and Q are not used, and each digit or letter may be used more than once. How many license plates are possible?

With your shoulder partner…USE YOUR BOARDS! A password is 4 letters followed by 1 digit. Uppercase letters (A) and lowercase letters (a) may be used and are considered different. How many passwords are possible?

On your own… Problem #7 A make-your-own adventure” story lets you choose 6 starting points, gives 4 plot choices, and then has 5 possible endings. How many adventures are there?

WHAT IF ORDER MATTERS? DO YOU THINK THIS CHANGES OUR CALCULATIONS?

VOCABULARY Permutation: a selection of a group of objects in which order is important. Combination: a selection of a group of objects in which order is not important.

Examples

PERMUTATIONS We use FACTORIALS to solve permutations Factorial: n!=n*(n-1)*(n-2)*…*1 Ex: 7!=7*6*5*4*3*2*1=5040 options

PERMUTATIONS-Problem #4 Suppose you want to order 5 people. How many different orders are possible?

PERMUTATIONS GENERAL NOTATION: n P r = We can think of this as:

PERMUTATIONS-Problem #5 You want to select and order 3 people from a group of 7. NOTATION: 7 P 3 =

PERMUTATIONS-Problem #6 We have 13 students and we want to figure out how many ways we could line 5 students up.

PERMUTATIONS---Problem #7 How many ways can a club select a president, vice president, and secretary from a group of 5?

PERMUTATIONS—Problem #8 With your shoulder partner An art gallery has 9 paintings from an artist and will display 4 from left to right along a wall. In how many ways can the gallery select and display 4 of the paintings?

PERMUTATIONS—Problem #9 With your shoulder partner Awards are given out at a Halloween party. How many ways can “most creative,” “silliest,” and “best” be awarded to 8 contestants if no one receives more than 1 award?

PERMUTATIONS—Problem #10 On your own How many ways can a 2-digit number be formed using only digits 5 through 9 (5, 6, 7, 8, and 9) and each digit being used only once?

COMBINATIONS Combination: a grouping of items in which order does not matter. Ex: You have 3 items: A, B, and C – How many permutations are there? – How many combinations?

NOTATION GENERAL NOTATION: n C r =

PERMUTATIONS vs. COMBINATIONS If order matters  PERMUTATION If order does not matter  COMBINATION

COMBINATIONS Problem #11: Katie is going to adopt kittens from a litter of 11. How many ways can she choose a group of 3 kittens? – DOES ORDER MATTER? 11 C 3 =

COMBINATIONS: With your shoulder partner Problem #12The swim team has 8 swimmers. Two swimmers will be selected to swim in the first heat. How many ways can the swimmers be selected?

STOP & THINK… 1.) Give a situation in which order matters and one in which order does not matter. 2.) Give the value of n C n where n is any integer. 3.) Tell what 3 C 4 would mean in the real world and why it is not possible.

GRAPHIC ORGANIZER TIME!