1. What is a permutation? A permutation is when you take a group of objects or symbols and rearrange them into different orders Examples: Four friends get in line to buy tickets to the dance. How many different ways could they line up? How many different ways could you arrange the letters in the word SCHOOL? Look for these key words: Arrange, order, sequence, line up, & permutation Objective 4: Probability How do I know if it is a permutation problem?

2. Movie Theater Example! Objective 4: Probability

3. How do I solve a permutation problem? Example: Four people get in line to buy lunch. How many different ways could they line up? Objective 4: Probability

4. How do I solve a permutation problem? Always start by making a spot for each position: Since there are 4 people in line, write out 4 spots ___ ___ Next, ask yourself how many different choices there are to fill the first spot  4 4 ___ ___ ___ Objective 4: Probability

5. How do I solve a permutation problem? Continue filling out each spot with the number of choices available If there were 4 choices for the 1 st spot, how many choices are left for the 2 nd spot  3 4 3 ___ ___ There must be 2 choices left for the 3 rd spot and only 1 choice for the last spot 4 3 2 1 _ Objective 4: Probability

6. How do I solve a permutation problem? Finally, multiply the number of choices in each spot together. 4 x 3 x 2 x 1 _ = 24 There are 24 different ways that 4 people can stand in line. Objective 4: Probability

7. More examples How many different ways can you arrange the letters in the word BOOKS? 1)Write out your spots ___ ___ ___ ___ ___ 2) Fill in the number of choices for each spot 5 4 3 2 1_ 3) Multiply 5 x 4 x 3 x 2 x 1 = 120 different ways to arrange the letters in BOOKS Objective 4: Probability

8. More examples 10 kids are running a race. How many different ways can they finish 1 st, 2 nd, and 3 rd ? 1)Write out your spots ___ ___ ___ 2) Fill in the number of choices for each spot 10 9 8_ 3) Multiply 10 x 9 x 8 = 720 different ways for the top three runners to finish. Objective 4: Probability

9. What is a combination? A combination is a different mixture of things where the order is not important. Think about this: ABC ACB BCA BAC CBA CAB Objective 4: Probability This list shows all the different permutations of the letters A, B, & C There is only 1 combination of letters here!!

10. Basketball Team Example! Objective 4: Probability

11. Permutation or Combination? 1.Choosing a baseball team from 36 boys 2. Deciding which of the boys will bat first, second, third, etc. 3. Deciding which three books to borrow from the library 4. Figuring out how many different ways two class presidents can be selected 5. Figuring out how many different ways a class president and vice president can be chosen Objective 4: Probability Combination Permutation Combination Permutation

12. What is factorial? Factorial means taking a number and multiplying it by all the numbers below it, until you get to 1 Example: 5 factorial means 5 x 4 x 3 x 2 x 1 = 120 3 factorial means 3 x 2 x 1 = 6 8 factorial means 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 = 40,320 Yes, the symbol is an exclamation point  ! 4! = 4 x 3 x 2 x 1 = 24 2! = 2 x 1 = 2 Objective 4: Probability Is there a symbol for factorial?

13. How do I solve combination problems? 1) Start off by treating it just like a permutation. 2) Draw your spots, fill them in, and multiply. 3) Then, divide by the number of spots factorial! Example: How many ways can a coach pick 5 starting players out of 9 girls on the basketball team? 1)Draw your 5 spots ___ ___ ___ ___ ___ 2)Fill them in and multiply: 9 x 8 x 7 x 6 x 5 = 15,120 3) Since there were 5 spots, divide by 5! 15,120 ÷ 5!  15,120 ÷ 120 = 126 different ways Objective 4: Probability

14. More Examples 1) Start off by treating it just like a permutation. 2) Draw your spots, fill them in, and multiply. 3) Then, divide by the number of spots factorial! Example: Eight people arrive to take a boat across the river. The captain says he can only take groups of four. How many different groups of 4 can be made? 1)Draw your 4 spots ___ ___ ___ ___ 2)Fill them in and multiply: 8 x 7 x 6 x 5= 1,680 3) Since there were 4 spots, divide by 4! 1,680 ÷ 4!  1,680 ÷ 24 = 70 different groups Objective 4: Probability

15. More Examples Example: How many different groups of 3 students can a teacher choose if there are 10 students in the class? 1)Draw your 3 spots ___ ___ ___ 2)Fill them in and multiply: 10 x 9 x 8 = 720 3) Since there were 3 spots, divide by 3! 720 ÷ 3!  720 ÷ 6 = 120 different groups Objective 4: Probability

16. More Examples Real-life Example: If there are 25 students in this class, and Mr. Z wants to have them sit at tables of 5, how many different table groups could he choose? 1)Draw your 5 spots ___ ___ ___ ___ ___ 2)Fill them in and multiply: 25 x 24 x 23 x 22 x 21 = 6,375,600 3) Since there were 5 spots, divide by 5! 6,375,600 ÷ 5!  6,37005,6 ÷ 120 = 53,130 different groups Objective 4: Probability