1. Do Now A password must contain 4 characters. 1. How many passwords are possible if you just use digits in the password? 2. How many are possible with just letters? 3. How many are possible if you can use both numbers and letters? 4. How many are possible if you use letters that are case-sensitive (upper and lower case are different) and the numbers?

2. Warm-up 1. 10 x 10 x 10 x 10 = 10,000 2. 26 x 26 x 26 x 26 = 456,976 3. 36 x 36 x 36 x 36 = 1,679,616 4. 62 x 62 x 62 x 62 =14,776,336

3. Section 10.2 Permutations LEQ: How do we solve problems involving permutations?

4. Permutations Permutation – an arrangement of objects in a specific order Linear permutation – where the objects are arranged in a row. (Assume all permutations are linear unless noted otherwise.)

5. Factorial (n!) = n x (n -1) x (n -2)…x 2 x 1. 6! = 6 x 5 x 4 x 3 x 2 x 1 = 720 Permutations of n Objects = n!

6. Permutations Ex. 1 A baseball lineup requires 9 players. How many possible lineups can be made using the same 9 players? 9! 362,880 different lineups.

7. Permutations Ex. 2 Find the number of ways to arrange the letters in the state MAINE. 5! 120 ways

8. Permutations Permutations of n Objects Taken r at a Time Notation = n P r or P(n,r) Can be found on the calculator or by using the formula:

9. Permutations Find the number of ways to listen to 4 CD’s from a selection of 8 CD’s. We have 8 objects (n) and are taking 4 (r)

10. Permutations Find the number of permutations of the first 15 letters of the alphabet, taking 5 letters at a time. We have 15 objects (n) and are taking 5 (r)

11. Permutations Find the number of ways to arrange the letters in the state OHIO. This will need to be done differently than before since we have repeating objects.

12. Permutations Permutations with Identical Objects If we have n objects with r identical objects, the number of permutations can be found by taking If we have multiple identical objects, we need to divide by each of them.

13. Permutations So for OHIO, we would need to do Find the number of ways we can arrange the letters in the state PENNSYLVANIA.

14. Circular Permutations Circular permutations – permutations that are arranged in a circle Can be found by: (n – 1)!

15. Circular Permutations Example How many different ways can 7 different items be arranged on a circular tray? (7 – 1)! 6! 720 ways

16. Homework Pg 640 – 641 Problems 10–46 even, 48,49,50,53,61,62,63

17. Solutions 10. 69640. 720 12. 242. 360 14. 21044. 10,080 16. 72046. 34,650 18. 210 20. 100048. 24 22. 0.0749. 720 24. 43,243,20050. 24 26. 33653. 5,040 28. 20,16061. 210 30. 40,32062. 720,720 32. 362,88063. 120 34. 259,459,200 36. 96,909,120 38. 427,518,000