Warm Up Which of the following are combinations? • 4 books pulled at random from a shelf • 12 books arranged on a shelf • 3 flavors of juice selected from a variety pack
Answer Which of the following are combinations? • 4 books pulled at random from a shelf • 12 books arranged on a shelf • 3 flavors of juice selected from a variety pack
With permutations, ORDER MATTERS! Which of the following situations would annoy you? You win a race, and your friend comes in second place. You receive the silver medal, and your friend gets the gold. When you ask why, you’re told that you and your friend were the top two, so why does it matter? You order a pizza with pepperoni and mushrooms. You watch the chef put the mushrooms on, and then the pepperoni. What the heck? You ask for pepperoni and mushrooms, not mushrooms and pepperoni!
Permutations • ORDER MATTERS! • Two ways to solve: • Fundamental Counting Principle • This formula: rselected items from a set of nitems • P =
Example 1 • In how many different orders can ten dogs line up to be groomed? • You can use the Fundamental Counting Principle or the formula!
Example 2: Your Turn! • In how many ways can you arrange six trophies on a shelf? • In how many ways can four tires be arranged on a car? • If the spare tire is included, how many ways can the tires be arranged on a car?
Sometimes, there isn’t a spot for every item! • For example, not every Olympian can get a medal. • That’s okay. We handle it the same way.
Example 3 • Seven yachts enter a race. • First, second and third places will be given to the three fastest yachts. • How many arrangements of first, second, and third places are possible with seven yachts?
Example 4: Your Turn! • Fifteen applicants want to interview with SAS. In how many ways can the 10 time slots be assigned? • How many different nine-player batting orders can be chosen from a baseball squad of 16? • There are 10 finalists in an archery competition. How many ways can the gold, silver & bronze medals be awarded?
Permutations with Repetition • Sometimes there are duplicate items in the set we are choosing from. • Ex. In the word LOLLIPOP, there are three “L,” two “O,” and two “P” items in the set. • The number of permutations of n items of which p are alike and q are alike is:
Example 5 • How many different ways can the letters of GEOMETRY be arranged? • Since there are two E’s, repetition is a factor.
Example 6 • How many different ways can the letters of the word MISSISSIPPI be arranged? • There are a total of 11 letters
Example 7 • How many different ways can the letters of the word MATHEMATICS be arranged? • There are a total of 11 letters
Permutations vs. Combinations Order doesn’t matter Key Words Choose Select Pick Use nCr • Order matters • Key Words • Arrange • Line up • Order • Use factorials, nPr, or multiplication counting principle without repeats with repeats
Permutation or Combination? A permutation is an ice cream cone…. but a combination is an ice cream bowl!
Example 8: Try these! • There is a total of 50 students in the junior class. How many ways can a class president, class vice president, class treasurer, and class secretary be chosen if each student may only hold one office? • The same class of 50 students wants to form a prom committee. How many ways can a four person prom committee be selected from the junior class? • If we want to form a group of five students, and we have 20 students to choose from, how many ways is this possible?
Example 9: Here’s a challenge! • How many ways can we arrange four letters from the word “computer” if repetitions are not allowed? • How many different four digit numbers are possible if we can choose any digits from 0 to 9 and all of the digits must be different?