6-7 Permutations and Combinations Hubarth Algebra II

A permutation is an arrangement of items in a particular order A permutation is an arrangement of items in a particular order. You can often find the permutations by using the Multiplication Counting Principle. Suppose you want to find the number of permutations for three items. There are three ways to choose the first item, two ways to choose the second item, and only one way to choose the third item. By the Multiplication Counting Principle, there are 3∙2∙1 permutations. Using factorial notation, you can write 3∙2∙1 as 3!, read “three factorial”

Ex. 1 Finding Permutations 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.

Ex. 2 Real-World Connection How many 4-letter codes 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 codes with no duplicates.

A selection in which order does not matter is a combination. Ex. 3 Finding Combinations Evaluate 10C4. 10C4 = 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

Ex. 4 Real-World Connection A disk jockey wants to select 5 songs from a new CD 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: n C r = 12 C 5 Use the nCr feature of your calculator. You can choose five songs in 792 ways.

Ex. 5 Real-World Connection 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.

Practice 1. In how many ways can you arrange nine trophies on a shelf? 9∙8∙7∙6∙……., 9! = 362880 2. How many arrangements of first, second , and third places are possible with 10 yachts? 10∙9∙8=720 10P3= 10! 10−3 ! = 10! 7! =720 3. Evaluate each expression. a. 10C5 b. 8C2 c. 25C7 252 28 480700 4. Of 20 books, in how many ways can you choose seven books? Twelve books? 77520; 125970