Bell Work. Get out the two copies of your desmos picture, the one copy of your equations, and the construction paper you brought.

Transcription

1 Bell Work Get out the two copies of your desmos picture, the one copy of your equations, and the construction paper you brought.

2 Introduction 1. List all the ways three different people can be standing in order.

3 Introduction 2. List all the ways four different runners can finish a race.

4 Bell Work 1. Vote for your top 10 projects on the back wall. Write the project # on each line. (#10 is the lowest of your top ten and #1 is the best) 2. If there is a runner in each of the eight lanes, how many different ways can the runners finish?

5 Discrete vs. Continuous A point has no length/width but an interval of points does have length. Chapter 9 is a collection of topics included under the branch of mathematics called Discrete Mathematics.

6 9.1 Basic Combinatorics Scientific studies will usually manipulate one or more explanatory variables and observe the effect on one or more response variables. The key to understanding the significance of the effect is to know what is likely to occur by chance alone. Doing this can depend on counting.

7 The Multiplication Principle of Counting If a procedure P has a sequence of stages S 1,S 2, S n and if S 1 can occur in r 1 ways, S 2 can occur in r 2 ways, S n can occur in r n ways, Then the number of ways that procedure P can occur is the product r 1 r 2 r n.

8 Example You have 3 pairs of khaki pants and 5 polos. How many different outfit options do you have?

9 Example Say a license plate consists of three letters of the alphabet followed by three numerical digits (0-9). Find the number of difference license plates that could be formed (a) If there is no restriction on the letters or digits that can be used (b) If no letter or digit can be repeated

10 Permutations One important application of the Multiplication Principle of Counting is to count the number of ways that a set of n objects (called an n-set) can be arranged in order. Each such ordering is called a permutation of the set.

11 Permutations If you have n objects There are n options for the first position There are n-1 options for the second position There are n-2 options for the third position The number of ways they can be ordered is n(n 1)(n 2) (2)(1).

12 Permutations There are n! permutations of an n-set.

13 Example Count the number of different 9-letter words that can be formed using the letters in each word. (a) DRAGONFLY

14 Example (b) BUTTERFLY (c) BUMBLEBEE

15 Distinguishable Permutations There are n! distinguishable permutations of a n-set containing n distinguishable objects. If an n-set contains n 1 objects of a first kind, n 2 objects of a second kind, and so on, with n 1 + n n k = n, then the number of distinguishable permutations of the n-set is n! n 1! n 2! n 3! n k!

16 Permutations of n objects taken r at a time Using n objects to fill r blanks in order. First blank has n options Second blank has (n 1) options Third blank has (n 2) options rth blank has n (r 1) options

17 Permutations of n objects taken r at a time By the multiplication principle the number of orders is n(n 1)(n 2) (n r + 1) This can be written as n! n r!

18 Permutation Counting Principle The number of permutations of n objects taken r at a time is denoted npr and is given by npr = n! n r! for 0 r n. If r > n, then npr = 0.

19 Example Evaluate without a calculator. Then confirm (a) and (b) with a calculator. (a) 6 P 4 (b) 11 P 3 (c) n P 3

20 Example Ten actors try out for roles as the two Von Trapp boys in the Sound of Music. Twenty actresses try out for roles as the five Von Trapp girls. In how many different ways can the director cast the seven roles?

21 9.1 (pg. 649) #1-16 Homework