}, how many different strings of length n 1 exist? }, how many different strings of length n 2 exist that contain at least one a 1

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1 1. [5] Give sets A ad B, each of cardiality 1, how may fuctios map A i a oe-tooe fashio oto B? 2. [5] a. Give the set of r symbols { a 1, a 2,..., a r }, how may differet strigs of legth 1 exist? [5]b. Give the set of r symbols { a 1, a 2,..., a r }, how may differet strigs of legth 2 exist that cotai at least oe a 1? 3. [10] Preset a combiatorial argumet that for all positive itegers x ad y k x k y k x y k = ( + ). = 0 (Hit: Cosider sequeces draw from the uio of distict sets A ad B of cardialities x ad y, respectively.) 4. [5] Give m a s, b s, ad p c s, how may distict sequeces are there that employ each of the m+ + p symbols exactly oce? 5. [5] If four six-sided dice are throw, how may differet cofiguratios (igorig order) are possible? (Uderstad: the cofiguratio {2, 2, 3, 4} is the same as the cofiguratio {2, 3, 2, 4 } but differet from {2, 3, 3, 4}) 6. [10] A pair of six-sided dice are throw r times. (We igore order i the pair but ot i the sequece. Thus the sequece begiig (1,2), (2,6), (1,3),... is the same as the sequece (2,1), (2,6), (1,3),... but differet from the sequece (2,6), (1,2), (1,3),...). How may such sequeces have each of (1,1), (2,2),..., ad (6,6) appearig at least oce withi the sequece? (Hit: First figure how may cofiguratios ca be displayed i a sigle throw of the pair. Secod, figure how may total sequeces of legth r there are, without cosiderig the requiremet for the 6 certai pairs. Lastly, let A i = the set of such sequeces avoidig (i,i) ad apply the iclusio/exclusio priciple.) 7. [5] a. Suppose k umbers are draw from {1, 2,..., } allowig repetitios. Cosiderig order to be relevat, how may such strigs are there? [5] b. Now assume each of the differet strigs i part a. is equally likely. What is the probability that the maximum of the k umbers is greater tha or equal to r, where 1 r? 8. [5] Cosider the followig tables of probabilities for gettig certai grades i a course ad beig i the freshma class. What is the probability of gettig a A give that a studet is a freshma? 9. [10] Suppose a message cosists of 0 s ad 1 s beig trasmitted with equal Freshma A.3.2 Less tha A.4.1 No-Freshma

2 likelihood util five 1 s have bee trasmitted (i.e., the message termiates with the fifth 1). Give a expressio for the expected value of the umber of bits i the message? (Do t waste time tryig to simplify the expressio.) 1. [5] Give set A of cardiality r 1 ad set B, of cardiality 1, how may fuctios map A i a oe-to-oe fashio ito B? 2. [5] a. Give the set of r symbols { a 1, a 2,..., a r }, how may differet strigs of legth 1 exist? [5]b. Give the set of r symbols { a 1, a 2,..., a r }, how may differet strigs of legth 2 exist that cotai at least two a 1 s? 3. [10] Preset a combiatorial argumet that for all positive itegers 1 k m r : r m r r k m = k k. m k 4. [5] Give m a s, b s, p c s ad q d s, how may distict sequeces are there that employ each of the m+ + p + q symbols exactly oce? 5. Cosider sequeces of the form < r 1, r 2,..., r >, where the r i 0. For fixed positive values of r ad, how may such sequeces are there satisfyig r + r r = r [5] a. Suppose k ad k umbers are draw from {1, 2,..., } without allowig repetitios. Cosiderig order to be relevat, how may such strigs are there? [5] b. Now assume each of the differet strigs i part a. is equally likely. What is the probability that the miimum of the k umbers is less tha or equal to r, where 1 r? 7. [5] Cosider the followig tables of probabilities for gettig certai scores o a exam ad beig a CS major or ot. Is the evet of gettig at lease a score of 80 idepedet of the evet that the studet is a CS major? CS Major o-cs Major 90 to to to to to [10] Suppose a message cosists of a total of 0 s ad 1 s beig trasmitted with equal likelihood. Give a expressio for the expected value of the umber of 1 s mius the umber of 0 s i the message? (Do t waste time tryig to simplify the expressio.)

3 1. [5] a. Suppose all people either have oe word ames (e.g. "Cher"), two word ames (e.g., "Jack Sprat") or three word ames (e.g. "Richard Milhous Nixo"). How may differet sets of iitials ca people have? [5]b. Suppose all 10 digit telephoe umbers obeyed these rules: the first digit caot be 1. The secod digit must be 0 or 1 the other 8 digits ca be ad decimal digit. How may such 10 digit telephoe umbers are there? 2. [5] a. How may permutatios of a, b, c, d, e, ad f have b to the left of c? [5]b. How may permutatios of a, b, c, d, e, ad f have b to the left of c ad e to the left of f? 3. [5] a. Preset a combiatorial argumet that for all positive itegers, a, ad b(>a): k a k b a k b ( ) =. k = 0 [5] b. Preset a combiatorial argumet that for all positive itegers : 2 2 = [10] A alphabet A cotais exactly characters. A message cosists of a ordered array of elemets of A (permittig repetitios)? If for each positio i the message each character is equally likely, what is the probability that a message of legth does ot cotai all characters. 5. [10] Suppose m are cards are to be draw from a 52 card deck with repetitio possible. What is the probability that cards 1, 2,, m-1 are ot hearts but card m is a heart? 6. [10] Let E be a set of equally likely evets ad A ad B be subsets of E. Show that Pr( AB ) = 1 + Pr( A) / Pr( B) Pr( A B) / Pr( B) 7. [5] Cosider the followig tables of probabilities for gettig certai scores o a exam ad beig a CS major or ot. Is the evet of gettig at lease a score of 70 idepedet of the evet that the studet is a CS major? CS Major o-cs Major 90 to to to to to [10] Suppose a message cosists of a total of 0 s ad 1 s beig trasmitted with equal likelihood. What is the expected umber of 0's?

4 1. [10]. Give set A of cardiality r 1 ad set B, of cardiality 1, how may fuctios mappig A ito B are ot oe-to-oe? 2. [10] For 2, determie the umber of strigs of a's ad b's of legth that either begi with the strig ab or ed with the strig ba or both. (Do ot igore the cases of = 2, ad = 3.) 3. a. [10] Assumig 1 m A = a1, a2,, a m B = b 1, b 2,, b. How may strigs usig each of these symbols exactly oce have the symbols of A ad B occurrig i the order give (i.e. i the strig a i must occur to the left of a j if 1 i < j m ad similarly for B)?,, Let { } ad { } b. [10] How may strigs usig each of these symbols exactly oce have the symbols of just A occurrig i the order give but ot ecessarily B? 4. [10] Usig a combiatorial argumet, prove that for 1: 2 k = 2 k = 0 5. a.[5] For 1, assume all strigs of legth from the set {a, b, c} (allowig repetitio) are equally likely. What is the probability that such a strig has o a? b. [5] What is the probability that such a strig has o b give that it has o a? 6. [10] Give o-egative itegers 1, 2, 3, ad 4, how may distict strigs of legth are there that have exactly 1 1's, 2 2's, 3 3's, ad 4 4's? 7. [10] How may triples 1, 2, 3 of o-egative itegers 1, 2, ad 3 exist that satisfy = 10. (Hit: Thik about balls ad bis.) 8. [10] For 1, assume all strigs of legth from the set {a, b, c} (allowig repetitio) are equally likely. What is the expected umber of c's. 1. [10] For 3, cosider bit strigs of legth. How may such strigs begi with the substrig 111, ed with the substrig 111, or both? (Do ot igore = 3, 4, ad 5.) 2. a. [10] Preset a combiatorial argumet that for all positive values of : i 3 = i j i= 0 j= 0

5 b. [10] Preset a combiatorial argumet that for all m ad satisfyig 2 m, 2, ad m +1: = + + m m m 1 m 2 (Hit: Cosider A= B {} c { d}, where c d, c B, d B, ad # B =.) 3. a. [10] For 1, how may decimal umbers betwee 1 ad 10 1 cotai o 5's or 7's.? b. [10] For 1, how may decimal umbers betwee 1 ad 10 1 cotai at most oe 5 ad oe 7? 4. [10] For 1, how may ways ca you fid ordered triples (, i j, k ) so that i, j, ad k are o-egative ad their sum is? (Hit: Cosider balls ad bis.) 5. [10] For 1, assume all strigs of digits from { 01,,...,} 9 are equally likely. What is the expected umber of 9's i such a strig? 6. [10] For 3, what is the probability that a strig of legth of a's b's, c's, ad d's has three or more a's (assumig all such strigs are equally likely)? 7. [10] Cosider two cards draw from a 52 card deck ad assume all such draws are equally likely. Is the evet that a heart is draw as the first card idepedet of the evet that a spade is draw of the secod card? 1. [10] For 4, cosider a set A of cardiality. How may subsets of A are of cardiality less tha or equal to 3? 2. a. [10] Preset a combiatorial argumet that for all ad k satisfyig 1 ad k :! = k! ( k)! k b. [10] Preset a combiatorial argumet that for all positive values of : + k 2 = 1 2 k = 0 (Hit: Cosider Let k be the positio of the first 1 i a bit strig.) 3. a. [10] For 1, cosider strigs of legth from the set of characters {a, b, c, d, e, f}allowig repetitio. How may such strigs at most oe a ad oe b? b. [10] For 1, how may decimal umbers betwee 1 ad 10 1 cotai at most oe 5 ad oe 7? 1

6 4. [10] For m 1, i how may ways ca you place m balls ito boxes so that every box has at least oe ball? 5. [10] For 1, assume all strigs of digits from { 01,,...,} 9 are equally likely. What is the expected umber of 9's i such a strig? 6. [10] For 3, what is the probability that a strig of legth of a's b's, c's, ad d's has three or more a's (assumig all such strigs are equally likely)? 7. [10] Cosider two cards draw from a 52 card deck ad assume all such draws are equally likely. Is the evet that a heart is draw as the first card idepedet of the evet that a spade is draw of the secod card? 1. [5] Cosider itegers i the set {1, 2, 3,, 1000}. How may are divisible by either 4 or 10? 2. a. [10] Preset a combiatorial argumet that for all 1: k 2 = 3 k = 0 k b. [10] Preset a combiatorial argumet that for all oegative itegers p, s, ad satisfyig p + s p p + s = p s p + s p (Hit: Cosider choosig two subsets.) 3. [10] For 1, Let A = {1, 2,, 2}. How may subsets of A cotai exactly k1eve umbers ad k2 odd umbers? 4. [10] For 1, how may ordered triples ( 1, 2, 3) of o-egative umbers satisfy =? (Hit: thik about puttig balls ito bis.) 5. [10] For 1, assume all strigs of characters from { abcd,,, } are equally likely. What is the expected umber of a's i such a strig? 6. [10] Give a fiite evet space E (i which all evets are equally likely) ad subsets A ad B of E, show that Pr( A B) Pr( A) + Pr( B) [10] Cosider a 52 card deck of cards from which the ace of spades is removed resultig i a 51 card deck. Further, cosider two distict cards draw from the 51 card deck ad assume all such uordered draws are equally likely. Lastly cosider the probability of the

7 evet that both cards are hearts. Is it more likely that both cards are hearts if it is give that both cards are face cards (i.e., Kigs, Quees, or Jacks)? 8. [10] Let A be a set of cardiality p. Cosider ordered strigs of legth m usig the elemets of A. How may such strigs have the m th compoet a repetitio of oe of the precedig m-1? (Hit: Thik about the complemet ad thik about selectig the m th compoet first.) 1. [5] Suppose all rolls of a six-side die are equally likely. What is the probability the roll is a six give that it is ot oe? 2. a. [10] Preset a combiatorial argumet that for all 1: = 2 1 k = 1 k (Note: The summatio begis with k = 1.) b. [10] Preset a combiatorial argumet that for all itegers k ad satisfyig 3 k = k k k 1 k 2 k 3 (Hit: Cosider three special elemets.) 3. [10] How may distict permutatios are there of the letters i mississippi? 4. [10] A bi has 100 blue balls, 100 red balls, ad 100 gree. How may differet collectios ca I obtai usig 100 of these balls? (Balls of the same color are idistiguishable from oe aother but are distiguishable from balls of aother color. A collectio has o order to it.) 5. [10] Assume all strigs of legth five usig characters from { abcd,,, } are equally likely. What is the probability that there is a substrig abc i the strig? 6. [10] Suppose a umber k from {1, 2,, 100} is to be draw ad that all umbers are equally likely. Let A be the evet k is a power of two. Let B be the evet k is a iteger multiple of four. Prove either that the evets A ad B are statistically idepedet or that they are statistically depedet. 7. [10] For 1, cosider strigs of legth cotaiig 0 s ad 1 s but edig i a 1. Assume all such strigs are equally likely. What is the expected umber of 1 s i such a strig? 8. [10] For 1, how may strigs of legth employig the characters {a,b,c} have at least oe a?

8 1. [5] For 3, how may subsets of size 3 from { a 1, a2,, a} are there that either cotai a 1 or a 2 (or both)? 2. [10] Give a set A of m characters, for 2, cosider strigs of legth usig ay of the characters of A. How may such strigs begi ad ed with the same character? 3. [10] Preset a combiatorial argumet that for all positive itegers m,, ad r, satisfyig r mi{ m, } : m+ m r = k = 0 k r k. (Hit: Cosider selectig from two sets.) b. [10] Preset a combiatorial argumet that for all positive itegers : i i 3 = i= 0 j= 0 i j (Note: Be very specific about the roles of i ad j.) 4. [10] How may distict permutatios are there of the digits i ? 5. [10] Give r 1, i how may ways ca idetical balls be placed ito r distict bis such that each bi cotais at least oe ball? (Hit: Cosider strigs with balls ad special dividers.) 6. [10] For 1, cosider strigs of legth 2 0 s ad 1 s. Assumig all such strigs are equally likely, what is the probability that such a strig has a equal umber of 0 s ad 1 s? 7. a. [10] For 5, cosider strigs of legth usig elemets of { abcd,,, }. Assume all such strigs are equally likely. What is the probability that a strig has exactly two a s? b. [5] What is the probability that such a strig has exactly three b s give that it has exactly two a s?