UNIT PROBABILITY 6 7 PERMUTATIONS AND COMBINATIONS Essential

UNIT: PROBABILITY 6 -7: PERMUTATIONS AND COMBINATIONS Essential Question: How is a combination different from a permutation?

6 -7: COMBINATIONS AND PERMUTATIONS Permutation: an arrangement of items in a particular order When all items of a particular set are used, you can alternatively use factorial notation. Example: In how many different ways can ten dogs line up to be groomed? Answer: There are 10 dogs to choose from first, multiplied by 9 dogs remaining to choose second, followed by 8 dogs to choose from to go third, etc. 10 • 9 • 8 • 7 • 6 • 5 • 4 • 3 • 2 • 1 = 10! = 3, 628, 800 ways Your turn: In how many ways can you line up 6 trophies on a shelf? 720 ways

6 -7: COMBINATIONS AND PERMUTATIONS Sometimes, not all items will be used. In this case, we can use the formula for permutations. Example: Seven yachts enter a race. First, second, and third places will be given. How many arrangements of 1 st, 2 nd, and 3 rd places are possible for the seven yachts? Answer: 7 possible 1 st place finishers, 6 remaining 2 nd place finishers, 5 possible 3 rd place finishers, means 7 • 6 • 5 = 210 arrangements Your turn: How many possible 1 st, 2 nd & 3 rd place arrangements are possible with 10 yachts? 720

6 -7: PERMUTATIONS AND COMBINATIONS When the order doesn’t matter, we use combinations The only difference (mathematically) between n. Cr and n. Pr is the addition of r! in the denominator Example: Evaluate

6 -7: PERMUTATIONS AND COMBINATIONS Example #4: A reading list for a course in world literature has 20 books on it. In how many ways can you choose four books to read? Answer: 20 books, choosing 4 = 20 C 4 = 4845 Your turn Evaluate 10 C 5 Of 20 books, in 252 how many ways can you choose seven books? 77, 520 ways

6 -7: PERMUTATIONS AND COMBINATIONS Example: Ten candidates are running for three seats in the student government. You may vote for as many as three candidates. In how many ways can you vote for three or fewer candidates? Answer: If you vote for three people, it’s 10 C 3 = 120 ways If you vote for two people, it’s 10 C 2 = 45 ways If you vote for on person, it’s 10 C 1 = 10 ways If you vote for no one, it’s 10 C 0 = 1 way 120 + 45 + 10 + 1 = 176 different ways Your turn: In how many ways can you vote for five or fewer people? 638 ways

6 -7: PERMUTATIONS AND COMBINATIONS Worksheet Problems 1 – 27 Odd problems