Permutations and Combinations 9.6 Pre-Algebra

Warm Up Find the number of possible outcomes. 1. bagels: plain, egg, wheat, onion meat: turkey, ham, roast beef, tuna 2. eggs: scrambled, over easy, hard boiled meat: sausage patty, sausage link, bacon, ham 3. How many different 4–digit phone extensions are possible? 16 12 10,000

Learn to find permutations and combinations.

Vocabulary factorial permutation combination

The factorial of a number is the product of all the whole numbers from the number down to 1. The factorial of 0 is defined to be 1. 5! = 5 • 4 • 3 • 2 • 1 Read 5! as “five factorial.” Reading Math

Examples: Evaluating Expressions Containing Factorials Evaluate each expression. A. 8! 8 • 7 • 6 • 5 • 4 • 3 • 2 • 1 = 40,320 8! B. 6! Write out each factorial and simplify. 8 •7 • 6 • 5 • 4 • 3 • 2 • 1 6 • 5 • 4 • 3 • 2 • 1 Multiply remaining factors. 8 • 7 = 56

Example: Evaluating Expressions Containing Factorials 10! (9 – 2)! C. 10! 7! Subtract within parentheses. 10 • 9 • 8 • 7 • 6 • 5 • 4 • 3 • 2 • 1 7  6  5  4  3  2  1 10 • 9 • 8 = 720

Try This Evaluate each expression. A. 10! 10 • 9 • 8 • 7 • 6 • 5 • 4 • 3 • 2 • 1 = 3,628,800 7! B. 5! Write out each factorial and simplify. 7 • 6 • 5 • 4 • 3 • 2 • 1 5 • 4 • 3 • 2 • 1 Multiply remaining factors. 7 • 6 = 42

Try This 9! C. (8 – 2)! 9! Subtract within parentheses. 6! 9 • 8 • 7 • 6 • 5 • 4 • 3 • 2 • 1 6  5  4  3  2  1 9 • 8 • 7 = 504

A permutation is an arrangement of things in a certain order. If no letter can be used more than once, there are 6 permutations of the first 3 letters of the alphabet: ABC, ACB, BAC, BCA, CAB, and CBA. first letter ? second letter ? third letter ? 3 choices 2 choices 1 choice • • The product can be written as a factorial. 3 • 2 • 1 = 3! = 6

Notice that the product can be written as a quotient of factorials. If no letter can be used more than once, there are 60 permutations of the first 5 letters of the alphabet, when taken 3 at a time: ABE, ACD, ACE, ADB, ADC, ADE, and so on. first letter ? second letter ? third letter ? 5 choices 4 choices 3 choices   = 60 permutations Notice that the product can be written as a quotient of factorials. 5 • 4 • 3 • 2 • 1 2 • 1 = 5! 2! 60 = 5 • 4 • 3 =

Example: Finding Permutations Jim has 6 different books. A. Find the number of orders in which the 6 books can be arranged on a shelf. The number of books is 6. 6! (6 – 6)! = 6! 0! = 6 • 5 • 4 • 3 • 2 • 1 1 = 6P6 = 720 The books are arranged 6 at a time. There are 720 permutations. This means there are 720 orders in which the 6 books can be arranged on the shelf.

Example: Finding Permutations B. If the shelf has room for only 3 of the books, find the number of ways 3 of the 6 books can be arranged. The number of books is 6. 6! (6 – 3)! = 6! 3! = 6 • 5 • 4 • 3 • 2 • 1 3 • 2 • 1 = 6P3 = 6 • 5 • 4 The books are arranged 3 at a time. = 120 There are 120 permutations. This means that 3 of the 6 books can be arranged in 120 ways.

Try This There are 7 soup cans in the pantry. A. Find the number of orders in which all 7 soup cans can be arranged on a shelf. The number of cans is 7. 7! (7 – 7)! = 7! 0! = 7 • 6 • 5 • 4 • 3 • 2 • 1 1 7P7 = = 5040 The cans are arranged 7 at a time. There are 5040 orders in which to arrange 7 soup cans.

Try This There are 7 soup cans in the pantry. B. If the shelf has only enough room for 4 cans, find the number of ways 4 of the 7 cans can be arranged. The number of cans is 7. 7! (7 – 4)! = 7! 3! = 7 • 6 • 5 • 4 • 3 • 2 • 1 3 • 2 • 1 7P4 = The cans are arranged 4 at a time. = 7 • 6 • 5 • 4 = 840 There are 840 permutations. This means that the 7 cans can be arranged in the 4 spaces in 840 ways.

A combination is a selection of things in any order.

If no letter is used more than once, there is only 1 combination of the first 3 letters of the alphabet. ABC, ACB, BAC, BCA, CAB, and CBA are considered to be the same combination of A, B, and C because the order does not matter. If no letter is used more than once, there are 10 combinations of the first 5 letters of the alphabet, when taken 3 at a time. To see this, look at the list of permutations on the next slide.

ABC ABD ABE ACD ACE ADE BCD BCE BDE CDE ACB ADB AEB ADC AEC AED BDC BEC BED CED BAC BAD BAE CAD CAE DAE CBD CBE DBE DCE BCA BDA BEA CDA CEA DEA DBC CEB DEB DEC CAB DAB EAB DAC EAC EAD DCB EBC EBD ECD CBA DBA EBA DCA ECA EDA DBC ECB EDB EDC These 6 permutations are all the same combination. In the list of 60 permutations, each combination is repeated 6 times. The number of combinations is = 10. 60 10

Example: Finding Combinations Mary wants to join a book club that offers a choice of 10 new books each month. A. If Mary wants to buy 2 books, find the number of different pairs she can buy. 10 possible books 10! 2!(10 – 2)! = 10! 2!8! 10C2 = 2 books chosen at a time 10 • 9 • 8 • 7 • 6 • 5 • 4 • 3 • 2 • 1 (2 • 1)(8 • 7 • 6 • 5 • 4 • 3 • 2 • 1) = = 45 There are 45 combinations. This means that Mary can buy 45 different pairs of books.

Example: Finding Combinations B. If Mary wants to buy 7 books, find the number of different sets of 7 books she can buy. 10 possible books 10! 7!(10 – 7)! = 10! 7!3! 10C7 = 7 books chosen at a time 10 • 9 • 8 • 7 • 6 • 5 • 4 • 3 • 2 • 1 (7 • 6 • 5 • 4 • 3 • 2 • 1)(3 • 2 • 1) = = 120 There are 120 combinations. This means that Mary can buy 120 different sets of 7 books.

Try This Harry wants to join a DVD club that offers a choice of 12 new DVDs each month. A. If Harry wants to buy 4 DVDs, find the number of different sets he can buy. 12 possible DVDs 12! 4!(12 – 4)! = 12! 4!8! 12C4 = 4 DVDs chosen at a time = 12 • 11 • 10 • 9 • 8 • 7 • 6 • 5 • 4 • 3 • 2 • 1 (4 • 3 • 2 • 1)(8 • 7 • 6 • 5 • 4 • 3 • 2 • 1) = 495

Try This; Continued There are 495 combinations. This means that Harry can buy 495 different sets of 4 DVDs.

Try This B. If Harry wants to buy 11 DVDs, find the number of different sets of 11 DVDs he can buy. 12 possible DVDs 12! 11!(12 – 11)! = 12! 11!1! 12C11 = 11 DVDs chosen at a time = 12 • 11 • 10 • 9 • 8 • 7 • 6 • 5 • 4 • 3 • 2 • 1 (11 • 10 • 9 • 8 • 7 • 6 • 5 • 4 • 3 • 2 • 1)(1) = 12

Try This; Continued There are 12 combinations. This means that Harry can buy 12 different sets of 11 DVDs.

Lesson Quiz Evaluate each expression. 1. 9! 2. 3. There are 8 hot air balloons in a race. In how many possible orders can all 8 hot air balloons finish the race? 4. A group of 12 people are forming a committee. How many different 4-person committees can be formed? 362,880 9! 5! 3024 40,320 495