CS 173: Discrete Mathematical Structures Dan Cranston dcransto@uiuc Siebel Center, rm 3240

1. CS 173:Discrete Mathematical Structures Dan Cranston dcransto@uiuc.edu Siebel Center, rm 3240 Office Hours: W,F 11:30a-12:30p

2. CS 173 Announcements • Homework 7 and 8 available. Due 3/12, 8a. Cs173 - Spring 2004

3. n-1Cj-1 n-1Cj CS 173 Pascal’s Identity A relationship between the entries in Pascal’s . Suppose T is a set, |T|=n. Let a be an element in T, and let S = T - {a}. Let’s count the nCj subsets of size j. Note that some of these contain a, and some don’t. How many contain a? How many don’t? Cs173 - Spring 2004

4. A m items B n items This is an exampleof a combinatorial proof. We describe the set counted by the LHS and RHS of the eqn, and argue that they count the same thing. Look in Rosen. CS 173 Vandermonde’s Identity Let m, n, and r be nonnegative integers with r not exceeding either m or n. Then To choose r items, take some from A and some from B. All possible ways of doing this gives the result. Cs173 - Spring 2004

5. CS 173 Combinations with repetition Suppose you want to buy 5 bags of chips from the 3 kinds you like at Meijer. In how many different ways can you stock up? Out of 7 items, we are choosing 2 to be bars. From that, and our understanding of the model, we can report the answer. Cs173 - Spring 2004

6. Example: How many solutions are there to the equation When the variables are nonnegative integers? 13C3 CS 173 Combinations with repetition There are n+r-1Cr, r-sized combinations from a set of n elements when repetition is allowed. Cs173 - Spring 2004

7. 6 3 CS 173 Permutations with indistinguishable objects How many different strings can be made from the letters in the word rat? How many different strings can be made from the letters in the word egg? Cs173 - Spring 2004

8. 8C4, now 4 spots are left 4C2, now 2 spots are left 2C2, now 0 spots are left CS 173 Permutations with indistinguishable objects How many different strings can be made from the letters in the phrase nano-nano? Key thoughts: 8 positions, 3 kinds of letters to place. In how many ways can we place the ns? In how many ways can we place the as? In how many ways can we place the os? Cs173 - Spring 2004

9. CS 173 Permutations with indistinguishable objects How many distinct permutations are there of the letters in the word APALACHICOLA? How many if the two Ls must appear together? How many if the first letter must be an A? Cs173 - Spring 2004

10. CS 173 A little practice A turtle begins at the upper left corner of an n x m grid and meanders to the lower right corner. How many routes could she take if she only moves right and down? Cs173 - Spring 2004

11. CS 173 A little practice A turtle begins at the upper left corner of a m x n grid and meanders to the lower right corner. How many routes could she take if she only moves right and down, and if she must pass through the dot at point (a,b)? Cs173 - Spring 2004

12. CS 173 A little practice In how many ways can 11 identical computer science books and 8 identical psychology books be distributed among 5 students? Hint: forget about the psychology books for the moment. Hint: how can you combine your soln for the CS books with your soln for the Psych books? Cs173 - Spring 2004

13. CS 173 A little practice In an RNA chain of 20 bases, there are 4 As, 5 Us, 6 Gs, and 5Cs. If the chain begins either AC or UG, how many such chains are there? Let A denote the set of chains beginning with AC, and U denote the set of chains beginning with UG. Count them separately, and then sum. First find |A|: 18 bases, 3 As, 5 Us, 6 Gs, and 4Cs. (This is like the MISSISSIPPI problem.) |A| = 18!/(3!5!6!4!) Cs173 - Spring 2004