6.7 – Permutations and Combinations

Permutations and Combinations Warm Up Simplify each expression. 1. 10 • 9 • 8 • 7 • 6 2.           3.  Let a b = 2a(a + b). Evaluate each expression. 4. 3 • 4 5. 2 • 7 6. 5 • 1 7. 6 • 10 4 • 3 • 2 6 • 5 7 • 6 • 5 • 4 • 3 • 2 • 1 4 • 3 • 2 • 1 *

Permutations and Combinations Solutions 1. 10 • 9 • 8 • 7 • 6 = 30,240 2. = 3. = 7 • 6 • 5 = 210 4. 3 • 4 with a • b = 2a(a + b): 2(3)(3 + 4) = 2(3)(7) = 6(7) = 42 5. 2 • 7 with a • b = 2a(a + b): 2(2)(2 + 7) = 2(2)(9) = 4(9) = 36 6. 5 • 1 with a • b = 2a(a + b): 2(5)(5 + 1) = 2(5)(6) = 10(6) = 60 7. 6 • 10 with a • b = 2a(a + b): 2(6)(6 + 10) = 2(6)(16) = 12(16) = 192 4 • 3 • 2 6 • 5 7 • 6 • 5 • 4 • 3 • 2 • 1 4 • 3 • 2 • 1 / / / 4 5 What does this have to do with permutations or combinations?

Permutations A permutation is an arrangement of items in a particular order. You can often find the number of permutations of some set of items using the Multiplication Counting Principle, or factorial notation n Factorial (n!) For any positive integer n, n! = n(n-1) * . . . * 3 * 2 * 1 For n=0, n!=1

Permutations and Combinations Example: In how many ways can 6 people line up from left to right for a group photo? Since everybody will be in the picture, you are using all the items from the original set. You can use the Multiplication Counting Principle or factorial notation. There are six ways to select the first person in line, five ways to select the next person, and so on. The total number of permutations is 6 • 5 • 4 • 3 • 2 • 1 = 6!. 6! = 720 The 6 people can line up in 720 different orders.

Let’s Try Some Find the following factorials: a. 12! b. -21! c. 5! 4! Is there a faster way to do this one? 5! = 5*4*3*2*1 4! = 4*3*2*1 479001600 -5.1 E 19 5

Permutations and Combinations The total number of permutations is 6 • 5 • 4 • 3 • 2 • 1 = 6!. How to use your graphing calculator to solve a factorial: Enter the number you are taking the factorial of (e.g. 6) Press MATH. 3. Cursor over to PRB for Probability 4. Press 4 to get ! option 5. Press ENTER. 6! = 720

Number of Permutations The number of permutations of n items of a set arranged r items at a time is n P r Where Example: 9P5 =

Permutations and Combinations Your new iPad requires a four-letter password. How many 4-letter passwords can be made if no letter can be used twice? Method 1:  Use the Multiplication Counting Principle. 26 • 25 • 24 • 23 = 358,800 Method 2:  Use the permutation formula. Since there are 26 letters arranged 4 at a time, n = 26 and r = 4. 26P4 = = = 358,800 26! (26 – 4)! 22! There are 358,800 possible arrangements of 4-letter passwords with no duplicates.

Let’s Try Some Evaluate the following: a) 3P2 b) 8P4 c) 5P3

Let’s Try Some Evaluate the following: a) 3P2 b) 8P4 c) 5P3

Combinations When the order of the items does not matter, we look for a combination. The number of n items of a set arranged r items at a time is n C r Where Example: 8C2 =

Combinations Evaluate 10C4. 10C4 = = / / / / / / = = = 210 10! 4!(10 – 4)! = 10! 4! • 6! 10 • 9 • 8 • 7 • 6 • 5 • 4 • 3 • 2 • 1 4 • 3 • 2 • 1 • 6 • 5 • 4 • 3 • 2 • 1 = / / / / / / 10 • 9 • 8 • 7 4 • 3 • 2 • 1 = = 210

Let’s Try Some Evaluate the following: a) 7C3 b) 25C2 c) 10C5

Let’s Try Some Evaluate the following: a) 7C3 b) 25C2 c) 10C5

Isn’t there an easier way??? Yes. But you need to be able to do the non-calculator method and theory too. Let’s start with the permutation 26P4 Enter the number if items in your set (e.g. 26) Press MATH. 3. Cursor over to PRB for Probability 4. Press 2 to get nPr option 5. Enter the arranged item set r (e.g. 4). 6. Press enter. 26P4 = 358800

Let’s Retry Our Problems from Earlier using the Permutation Function on the Graphing Calculator Evaluate the following: a) 3P2 b) 8P4 c) 5P3 Solutions: a) 6 b) 1680 c) 60

Combinations Write: nCr = 12C5 =792 A DJ wants to select 5 songs from a new album that contains 12 songs. How many 5-song selections are possible? Relate: 12 songs chosen 5 songs at a time Define: Let n = total number of songs. Let r = number of songs chosen at a time. Write: nCr = 12C5 =792 Similar to our permutation shortcut, you can use the combination shortcut You can choose five songs in 792 ways.

Combinations A pizza menu allows you to select 4 toppings at no extra charge from a list of 9 possible toppings. In how many ways can you select 4 or fewer toppings? You may choose  4 toppings, 3 toppings, 2 toppings, 1 toppings, or none. 9C4 9C3 9C2 9C1 9C0 The total number of ways to pick the toppings is 126 + 84 + 36 + 9 + 1 = 256. There are 256 ways to order your pizza.

Let’s Retry Our Problems from Earlier using the Combination Function on the Graphing Calculator Evaluate the following: a) 7C3 b) 25C2 c) 10C5 Solutions: a) 35 b) 480,700 c) 252

In Summary How can you decide if you have a permutation or a combination? Permutations use the entire group and order matters Ex: How many ways can you arrange all the students in the class into one line? Combinations use parts of the group and the order does not matter Ex: If you have 10 people in a group, how many one on one conversations can you have?