1. Some lovely and slightly random facts… Give the number of objects described. 1. The number of cards in a standard deck 52 2. The number of cards of each suit in a standard deck 13 3. The number of faces on a cubical die 6 4. The number of possible totals when two dice are rolled 11 5. The number of vertices of a decagon 10 6. The number of musicians in a string quartet 4

2. More lovely and slightly random facts… Give the number of objects described. 7. The number of players on a soccer team 11 8. The number of prime numbers between 1 and 10, inclusive 4 9. The number of squares on a chessboard 64 10. The number of cards in a contract bridge hand 13 11. The number of players on a rugby team 15

3. Chapter 9:1a Discrete Mathematics

4. What is Discrete Mathematics??? Any interval (a, b) contains a continuum of real numbers  you can “zoom in” forever and there will still be an interval there… In contrast, discrete mathematics involves separate (or “discrete”) numbers that do not lie on a continuum. The simplest type of discrete mathematics is… The mathematics of the continuum is used a great deal in algebra, analysis, and geometry, which all lead to important concepts in calculus… COUNTING!!!

5. The Importance of Counting In how many ways can three distinguishable objects be arranged in order? (let’s call them A, B, and C) We can “see” the solution to this problem with a tree diagram: Starting Point A B C B C C B A C C A A B B A The six different orderings: ABC, ACB, BAC, BCA, CAB, CBA

6. Multiplication Principle of Counting (our tree diagram gives a visualization of this principle…) If a procedure P has a sequence of stages S, S, …, S and if 1 2 n S can occur in r ways, 1 1 2 2 n n then the number of ways that the procedure P can occur is the product: r r r. 12n

7. Multiplication Principle of Counting Important note about this principle!!! Be mindful of how the choices at each stage are affected by the choices at preceding stages… (think about how this applies In our very first example)

8. Using the Multiplication Principle A certain state’s license plates consist of three letters followed by three numerical digits (0 through 9). Find the number of different license plates that could be formed (a) if there is no restriction on the letters or digits; (b) if no letter or digit can be repeated. With no restrictions, we have 26 ways to fill each of the first three blanks, and 10 ways to fill each of the final three blanks. Multiplication Principle: 26 x 26 x 26 x 10 x 10 x 10 = 17,576,000 ways

9. Using the Multiplication Principle A certain state’s license plates consist of three letters followed by three numerical digits (0 through 9). Find the number of different license plates that could be formed (a) if there is no restriction on the letters or digits; (b) if no letter or digit can be repeated. If no repeats are allowed, there are 26 choices for the first blank, 25 for the second, 24 for the third, 10 for the fourth, 9 for the fifth, and 8 for the sixth. Multiplication Principle: 26 x 25 x 24 x 10 x 9 x 8 = 11,232,000 ways Are these realistic estimates for #’s of ways???

10. Permutations

11. What are they??? Permutations – possible orderings of a set of objects Consider the  There are 24 possible orderings………………………can we generalize for a rule ? RULE: If there are n objects in a set, then there are n ! permutations of the set Note: the symbol n! (read “n factorial”) represents the product n(n – 1)(n – 2)(n – 3)…(2)(1). In addition, we define 0! = 1.

12. What are they??? Permutations – possible orderings of a set of objects RULE: If there are n objects in a set, then there are n ! permutations of the set This rule only works if all elements of a set are distinguishable from each other…

13. Distinguishable Permutations There are n! distinguishable permutations of an n-set containing n distinguishable objects. If an n-set contains n objects of a first kind, n objects of a second kind, and so on, with n + n + … + n = n, then the number of distinguishable permutations of the n-set is 12 12k

14. Some Practice Problems Count the number of different 9-letter “words” (don’t worry about whether they’re in the dictionary) that can be formed using the letters in each of the given words. 1. DRAGONFLY Each permutation of the 9 letters forms a different word. There are 9! = 362,880 such permutations. 2. BUTTERFLY There are also 9! permutations, but simply switching the two T’s does not result in a new word. There are = 181,440 distinguishable permutations. 9! 2!

15. Some Practice Problems Count the number of different 9-letter “words” (don’t worry about whether they’re in the dictionary) that can be formed using the letters in each of the given words. 3. BUMBLEBEE The three B’s and three E’s are indistinguishable, so we divide by 3! twice to correct for the overcount… 9! 3!3! There are = 10,080 distinguishable permutations.

16. Permutation Counting Formula In some counting problems, we are interested in using n objects to fill r blanks in order, where r < n. These are called permutations of n objects taken r at a time. The number of permutations of n objects taken r at a time is denoted P and is given by rn for If r > n, then What is ?

17. More Practice Problems Evaluate the given expressions, first without a calculator. Then, check your answer with a calculator. 1. 2.

18. More Practice Problems Evaluate the given expression. 3.

19. More Practice Problems Sixteen actors answer a casting call to try out for roles as dwarfs in a production of Snow White and the Seven Dwarfs. In how many ways can the director cast the seven roles? There are 7 blanks to be filled, and 16 actors with which to fill them: ways

20. More Practice Problems Proctor has 22 players train for his rugby team, and he must fill 15 playing spots for a weekend match. Assuming that all players are equally qualified for each playing position, in how many ways can Proctor fill the spots? ways