1. 15.3 Permutations and Combinations OBJ:  To solve problems involving permutations and combinations

2. Permutation (Order Important) Arrangement Picture EX:  A club with 12 members wants to choose a president, vice-president, and treasurer. The order of choices is important: The order A, B, and C for president, vice-president, and treasurer, respectively, is different from the order B, A, and C for the 3 positions. The number of ways of filling the three offices is: 12 11 10 = 1320 Each slate of officers is a permutation of 3 people from a set of 12, denoted as 12 P 3.

3. Combination (Order not impt.) Committee If the club only wanted to select a committee of 3 out of its 12 members, order is not important. Each of the following selections: ABC, ACB, BAC, BCA, CAB, CBA are the same. To determine the number of different committees of three that can be chosen you would need to divide the previous total by 6 12 11 10 1320 3 2 1 = 6 = 220 Each committee is a combination of 3 people from a set of 12, denoted as 12 C 3.

4. Permutation (order important) Example 12 P 3 = 12 11 10 12 11 10 9! (12 – 3) ! = 1320 Formula n P r = n (n-1)(n-2)... (n-r+1)(n-r)! (n – r)! = n!__ (n - r )!

5. Combination (order not important) Example 12 C 3 = 12 11 10 3 2 1 = 12 11 10 (12–3)! (12 – 3)! 3! = 220 Formula n C r = n (n-1)(n-2)... (n-r+1)(n-r)! (n-r)!1 2 3... r = n!__ ( n - r )! r!

6. EX:  A company advertises two job openings, one for a copywriter and one for an artist. If 10 people who are qualified for either position apply, in how many ways can the openings be filled? Fundamental Counting Principle 10 9 = 90 Permutation Formula 10 P 2 = 10! (10 – 2)! 10 9 8! 8! 10 9 = 90

7. EX:  A company advertises two job openings for computer programmers, both with the same salary and job description. In how many ways can the openings be filled if 10 people apply. Fundamental Counting Principle 10 9 2 1 = 90 2 = 45 Combination Formula 10 C 2 10! (10 – 2)! 2! 10 9 8! 8! 2! 10 9 2 1 = 90 2 = 45

8. EX:  For a raffle, 845 tickets are sold. a)In how many ways can four $50 gift certificates be awarded 845 C 4 845! (845-4)! 4! 845 844 843 842 4 3 2 1 2.1 10 10 21,000,000,000 b) In how many ways can a $100, a $50, a $20m and a $10 gift certificate be awarded? 845 P 4 845! (845-4)! 845 844 843 842 5.06 10 11 506,000,000,000

9. EX:  At the Red Lion Diner, an omelet can be ordered plain or with any or all of the following fillings: cheese, onions, and peppers. How many different kinds of omelets are possible? a) 3 C 3 3! __ (3 – 3)! 3! 1 b) 3 C 2 3! __ (3 – 2)! 2! 3 c) 3 C 1 3!_ _ (3 – 1)! 1! 3 d) 3 C 0 3!__ (3 – 0)! 0! 1 3 C 3 + 3 C 2 + 3 C 1 + 3 C 0 1 + 3 + 3 + 1 = 8