Discrete Mathematics- An Open Introduction - 3rd Edition, 2016a

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104 1. Counting What do outcomes actually look like? How can we represent them? One approach would be to write an outcome as a string of four numbers like this: 3112, which represent the outcome in which the first kid gets 3 cookies, the second and third kid each get 1 cookie, and the fourth kid gets 2 cookies. Represented this way, the order in which the numbers occur matters. 1312 is a different outcome, because the first kid gets a one cookie instead of 3. Each number in the string can be any integer between 0 and 7. But the answer is not 7 4 . We need the sum of the numbers to be 7. Another way we might represent outcomes is to write a string of seven letters: ABAADCD, which represents that the first cookie goes to kid A, the second cookie goes to kid B, the third and fourth cookies go to kid A, and so on. In fact, this outcome is identical to the previous one—A gets 3 cookies, B and C get 1 each and D gets 2. Each of the seven letters in the string can be any of the 4 possible letters (one for each kid), but the number of such strings is not 4 7 , because here order does not matter. In fact, another way to write the same outcome is AAABCDD. This will be the preferred representation of the outcome. Since we can write the letters in any order, we might as well write them in alphabetical order for the purposes of counting. So we will write all the A’s first, then all the B’s, and so on. Now think about how you could specify such an outcome. All we really need to do is say when to switch from one letter to the next. In terms of cookies, we need to say after how many cookies do we stop giving cookies to the first kid and start giving cookies to the second kid. And then after how many do we switch to the third kid? And after how many do we switch to the fourth? So yet another way to represent an outcome is like this: ∗ ∗ ∗| ∗ | ∗ | ∗ ∗. Three cookies go to the first kid, then we switch and give one cookie to the second kid, then switch, one to the third kid, switch, two to the fourth kid. Notice that we need 7 stars and 3 bars – one star for each cookie, and one bar for each switch between kids, so one fewer bars than there are kids (we don’t need to switch after the last kid – we are done). Why have we done all of this? Simple: to count the number of ways to distribute 7 cookies to 4 kids, all we need to do is count how many stars and bars charts there are. But a stars and bars chart is just a string of symbols,
1.5. Stars and Bars 105 some stars and some bars. If instead of stars and bars we would use 0’s and 1’s, it would just be a bit string. We know how to count those. Before we get too excited, we should make sure that really any string of (in our case) 7 stars and 3 bars corresponds to a different way to distribute cookies to kids. In particular consider a string like this: | ∗ ∗ ∗ || ∗ ∗ ∗ ∗. Does that correspond to a cookie distribution? Yes. It represents the distribution in which kid A gets 0 cookies (because we switch to kid B before any stars), kid B gets three cookies (three stars before the next bar), kid C gets 0 cookies (no stars before the next bar) and kid D gets the remaining 4 cookies. No matter how the stars and bars are arranged, we can distribute cookies in that way. Also, given any way to distribute cookies, we can represent that with a stars and bars chart. For example, the distribution in which kid A gets 6 cookies and kid B gets 1 cookie has the following chart: ∗ ∗ ∗ ∗ ∗ ∗ | ∗ ||. After all that work we are finally ready to count. Each way to distribute cookies corresponds to a stars and bars chart with 7 stars and 3 bars. So there are 10 symbols, and we must choose 3 of them to be bars. Thus: ( ) 10 There are ways to distribute 7 cookies to 4 kids. 3 While we are at it, we can also answer a related question: how many ways are there to distribute 7 cookies to 4 kids so that each kid gets at least one cookie? What can you say about the corresponding stars and bars charts? The charts must start and end with at least one star (so that kids A and D) get cookies, and also no two bars can be adjacent (so that kids B and C are not skipped). One way to assure this is to place bars only in the spaces between the stars. With 7 stars, there are 6 spots between the stars, so we must choose 3 of those 6 spots to fill with bars. Thus there are ( 6) 3 ways to distribute 7 cookies to 4 kids giving at least one cookie to each kid. Another (and more general) way to approach this modified problem is to first give each kid one cookie. Now the remaining 3 cookies can be distributed to the 4 kids without restrictions. So we have 3 stars and 3 bars for a total of 6 symbols, 3 of which must be bars. So again we see that there are ( 6 3) ways to distribute the cookies. Stars and bars can be used in counting problems other than kids and cookies. Here are a few examples:
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Discrete Mathematics An Open Introd
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Oscar Levin School of Mathematical
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Acknowledgements This book would no
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Preface This text aims to give an i
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How to use this book In addition to
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Contents Acknowledgements Preface H
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Contents xiii Proof by Cases . . .
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Chapter 0 Introduction and Prelimin
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0.1. What is Discrete Mathematics?
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0.2. Mathematical Statements 5 •
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0.2. Mathematical Statements 7 Impl
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0.2. Mathematical Statements 9 3. I
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0.2. Mathematical Statements 11 The
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0.2. Mathematical Statements 13 It
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0.2. Mathematical Statements 15 Inv
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0.2. Mathematical Statements 17 and
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0.2. Mathematical Statements 19 7.
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0.2. Mathematical Statements 21 12.
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0.2. Mathematical Statements 23 (b)
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0.3. Sets 25 say that the set B is
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0.3. Sets 27 We already have a lot
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0.3. Sets 29 Example 0.3.3 Let A {
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0.3. Sets 31 Operations On Sets Is
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0.3. Sets 33 You might notice that
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0.3. Sets 35 Exercises 1. Let A {1
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0.3. Sets 37 17. Let A {a, b, c, d
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0.4. Functions 39 0.4 Functions A f
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0.4. Functions 41 It would be absol
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0.4. Functions 43 We will also be i
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0.4. Functions 45 2. We are told th
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0.4. Functions 47 2. g : {1, 2, 3}
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0.4. Functions 49 Example 0.4.8 Con
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0.4. Functions 51 Exercises 1. Cons
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2.2. Arithmetic and Geometric Seque
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2.3. Polynomial Fitting 161 sequenc
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2.3. Polynomial Fitting 163 a 2 7
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2.3. Polynomial Fitting 165 5. Make
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2.4. Solving Recurrence Relations 1
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2.5. Induction 177 2.5 Induction Ma
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2.5. Induction 179 2. Prove that if
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2.5. Induction 181 the next is easi
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2.5. Induction 183 But we want to k
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2.5. Induction 185 Investigate! Str
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2.5. Induction 187 a − 1 breaks;
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2.5. Induction 189 8. Zombie Euler
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23. Use induction to prove that 2.5
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2.6. Chapter Summary 193 Investigat
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2.6. Chapter Summary 195 (c) Find a
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Chapter 3 Symbolic Logic and Proofs
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3.1. Propositional Logic 199 Truth
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3.1. Propositional Logic 201 statem
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3.1. Propositional Logic 203 propos
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3.1. Propositional Logic 205 How do
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3.1. Propositional Logic 207 Beyond
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3.1. Propositional Logic 209 Exerci
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3.1. Propositional Logic 211 ∴ P
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3.2. Proofs 213 Investigate! 3.2 Pr
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3.2. Proofs 215 5. N is not divisib
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3.2. Proofs 217 to prove directly,
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3.2. Proofs 219 Example 3.2.7 Prove
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3.2. Proofs 221 In fact, we can qui
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3.2. Proofs 223 Exercises 1. Consid
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3.2. Proofs 225 14. Prove that ther
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3.3. Chapter Summary 227 3.3 Chapte
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3.3. Chapter Summary 229 6. Conside
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Chapter 4 Graph Theory Investigate!
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4.1. Definitions 233 Investigate! 4
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4.1. Definitions 235 we have edges
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4.1. Definitions 237 Alternatively,
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4.1. Definitions 239 2-element subs
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4.1. Definitions 241 the vertices w
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4.1. Definitions 243 Path A path is
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4.1. Definitions 245 9. For each of
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4.2. Trees 247 Investigate! 4.2 Tre
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4.2. Trees 249 Assume T is a tree,
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4.2. Trees 251 one. Let T ′ be th
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4.2. Trees 253 are adjacent (they a
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4.2. Trees 255 Exercises 1. Which o
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4.2. Trees 257 a d b c f e 14. Give
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4.3. Planar Graphs 259 WARNING: you
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4.3. Planar Graphs 261 which says t
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4.3. Planar Graphs 263 sphere to li
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4.3. Planar Graphs 265 which will b
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4.4. Coloring 267 Investigate! 4.4
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4.4. Coloring 269 representations o
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4.4. Coloring 271 KQEJ KQEA KQEB P
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4.4. Coloring 273 We must color the
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4.4. Coloring 275 6. Prove the chro
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4.5. Euler Paths and Circuits 277 I
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4.5. Euler Paths and Circuits 279 W
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4.5. Euler Paths and Circuits 281 (
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4.6. Matching in Bipartite Graphs 2
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4.7. Chapter Summary 289 4.7 Chapte
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4.7. Chapter Summary 291 (b) For wh
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4.7. Chapter Summary 293 (d) Every
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Chapter 5 Additional Topics 5.1 Gen
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5.1. Generating Functions 297 shift
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5.1. Generating Functions 299 numbe
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5.1. Generating Functions 301 relat
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5.1. Generating Functions 303 and t
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5.1. Generating Functions 305 7. Us
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5.2. Introduction to Number Theory
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Appendix A Selected Hints 0.2 Exerc
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Selected Hints 327 1.6 Exercises 1.
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Selected Hints 329 2.5.27. For the
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Selected Hints 331 3.2.16. Your fri
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Selected Hints 333 4.4.10. The chro
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Appendix B Selected Solutions 0.2.1
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Selected Solutions 337 (d) Equivale
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Selected Solutions 339 (b) We get t
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Selected Solutions 341 (d) Such an
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Selected Solutions 343 Proof. Let x
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Selected Solutions 345 (c) 2 6 −
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Selected Solutions 347 (c) To build
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Selected Solutions 349 the box. The
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Selected Solutions 351 (c) ( 16) [(
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Selected Solutions 353 (p) Neither.
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Selected Solutions 355 x + y n. So
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Selected Solutions 357 2.1.4. (a) T
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Selected Solutions 359 (b) 6n + 1,
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Selected Solutions 361 2.4.12. We h
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Selected Solutions 363 F 2k+3 −
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Selected Solutions 365 2.6.7. (a) 4
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Selected Solutions 367 3.1.16. (a)
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Selected Solutions 369 P Q R P →
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Selected Solutions 371 (b) The conv
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Selected Solutions 373 (c) Not poss
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Selected Solutions 375 number of fa
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Selected Solutions 377 4.6 Exercise
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Selected Solutions 379 (b) Now addi
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Selected Solutions 381 4.7.21. (a)
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Selected Solutions 383 5.2.6. (a) 3
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Appendix C List of Symbols Symbol D
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Index additive principle, 57 adjace
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Index 389 Goldbach conjecture, 227
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Index 391 vs combination, 84, 123,
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Index 393 set difference, 34 vertex
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Colophon This book was authored in
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