SETS AND COUNTING PRINCIPLES

1) Describe what a set is and how sets are denoted.

2) Give examples of collections that are sets.

3) Give examples of collections that are not sets.

4) Describe the empty set, the associated symbols, and give examples of sets that are empty.

5) Describe subsets and give examples.

6) Define cardinality of a set, its symbol, and give examples.

7) Define power set and its cardinality.

8) Write the members of the power set of the sets (i.e., list all possible subsets of):††††††

†††† a) {A, B}†† b) {A, B, C}†† c) {A, B, C, D}†† d) {A, B, C, D, E}

9)Fundamental multiplication principle of mathematics

†††† a)AN EXAMPLE:Consider the sets E = {a, b, c} and F = {1, 2}.Write out all possible pairs of elements in such a way that the first element is from set E and the second element is from set F.Construct a tree diagram and a matrix.

 

††††† TREEDIAGRAM†††††††††††††††††††††††††††††††† ORDERED PAIRS

†† ††††††††††††††††††††††††††††††††††††††††††

††††††††† †††††††††††††††††††††††††††††††††††††††††††††††††

††††††††††††††††††††††††††††††††††††††††††††††††††††††††††

 

root†††††††††††††††††††††††††††††††††††††††††††††††††††††

†††††††††††††††††††††††††††††††††††††††††††††††††††††††††††

 


†††††††††††††††† †††††††††††††††††††††††††††††††††††††††††††

†††††††††††††††††††††††††††††††††††††††††††††††††††††††††††

 


†††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††† F

MATRIX

E

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

††† b) Statement of the Principle(for two successive events , )

 

10) A person wants to purchase a cellular phone and a calling plan.Suppose that there are two choices of cellular phones (the Motorola and the Nokia) and three choices of calling plans (one for $29.99 which allows 300 minutes of airtime per month, a second for $39.99 which allows 600 minutes of airtime per month, and a third for $49.99 which allows 1000 minutes of airtime per month).In how many different ways can this person purchase a cellular phone and a calling plan?Show all possibilities using a matrix and using a tree diagram.

 

11) A game consists of tossing a coin and then rolling a die.How many different outcomes are possible?Show all possible outcomes using a matrix and a tree diagram.

††††††† SOLUTION:Coin: {Heads, Tails}.Cardinality = 2

††††††† Die: {1, 2, 3, 4, 5, 6}.Cardinality = 6

††††††† Total = (2)(6)=12 possible outcomes.

 

††††††††† a) MATRIX††††††††††††††††††††††††††††††††††††††††††††††††††††††††† DIEOUTCOMES

COIN

1

2

3

4

5

6

Heads (H)

(H, 1)

(H, 2)

(H, 3)

(H, 4)

(H, 5)

(H, 6)

Tails (T)

(T, 1)

(T, 2)

(T, 3)

(T, 4)

(T, 5)

(T, 6)

 

††††††††† b) TREE DIAGRAM

††††††††††††††††††††††††††††††††††††††††††††††††††††††††† POSSIBLE OUTCOMES

 

††††††††††††††††††††††††††††††††††††††††††††††† 1†††††† (H, 1)

††††††††††††††††††††††††††††††††††††††††††††††† 2†††††† (H, 2)

††††††††††††††††††††††††††††††††††††††††††††††† 3†††††† (H, 3)

††††††††††††††††††† Heads†††††††††††††††††† 4†††††† (H, 4)

††††††††††††††††††††††††††††††††††††††††††††††† 5†††††† (H, 5)

††††††††††††††††††††††††††††††††††††††††††††††† 6†††††† (H, 6)

 

††††††††††††††††††††††††††††††††††††††††††††††† 1†††††† (T, 1)

††††††††††††††††††††††††††††††††††††††††††††††† 2†††††† (T, 2)

††††††††††††††††††††††††††††††††††††††††††††††† 3†††††† (T, 3)

†††††††††††††††††† Tails†††††††††† †††††††††† 4††††† (T, 4)

††††††††††††††††††††††††††††††††††††††††††††††† 5†††††† (T, 5)

††††††††††††††††††††††††††††††††††††††††††††††† 6†††††† (T, 6)

 

†††† In both cases, we can view the 12 possible outcomes.

 

12) A particular code consists of two digits, each from the set{0, 1, 2, 3, 4, 5, 6, 7, 8, 9}.How many different two-digit codes are possible?

 

13)State the general multiplication principle.

 

14) At a particular restaurant, customers can order a meal consisting of one choice of {steak, chicken, fish}, plus one choice of {baked potato, mashed potatoes}, plus one choice of {water, soda, juice}.In how many different ways can a customer order a meal?

††† SOLUTION

††† (3)(2)(3)=18 different ways.

 

 

 

TREE DIAGRAM

†††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††† †††† POSSIBLEMEALS

 

Water††††††††††† (Steak, Baked Potato, Water)

Baked Potato†††††††††††† Soda††††††††††††† (Steak, Baked Potato, Soda)

Juice††††††††††††† (Steak, Baked Potato, Juice)

††††††††††† Steak

Water††††††††††† (Steak, Mashed Potatoes, Water)

Mashed Potatoes††††† Soda††††††††††††† (Steak, Mashed Potatoes, Soda)

Juice††††††††††††† (Steak, Mashed Potatoes, Juice)

 

Water††††††††††† (Chicken, Baked Potato, Water)

Baked Potato†††††††††††† Soda††††††††††††† (Chicken, Baked Potato, Soda)

Juice††††††††††††† (Chicken, Baked Potato, Juice)

††††††††††† Chicken

Water††††††††††† (Chicken, Mashed Potatoes, Water)

Mashed Potatoes††††† Soda††††††††††††† (Chicken, Mashed Potatoes, Soda)

Juice††††††††††††† (Chicken, Mashed Potatoes, Juice)

 

Water††††††††††† (Fish, Baked Potato, Water)

Baked Potato ††††††††††† Soda††††††††††††† (Fish, Baked Potato, Soda)

Juice††††††††††††† (Fish, Baked Potato, Juice)

†††††††††††† Fish

Water††††††††††† (Fish, Mashed Potatoes, Water)

Mashed Potatoes††††† Soda††††††††††††† (Fish, Mashed Potatoes, Soda)

Juice††††††††††††† (Fish, Mashed Potatoes, Juice)

 

15) A certain model of vehicle is available in twelve different colors, four different styles {hatchback, sedan, SUV, or station wagon}, {manual or automatic transmission}, and {two-door or four-door}.How many different possible choices of this vehicle are there?

16) A pizza can be ordered with four choices of size {small, medium, large, super-large}, four choices of crust {thin, thick, crispy, regular}, and ten choices of toppings {extra cheese, beef, chicken, ham, bacon, sausage, pepperoni, mushrooms, onions, green peppers}.How many different one-topping pizzas can be ordered?

17) A restaurant offers five appetizers, six main courses, seven beverages, four desserts.If exactly one item is selected from each group, in how many ways can a person order a meal?

††††† ††† SOLUTION

††††† {appetizer}{main course}{beverage}{dessert} = (5)(6)(7)(4) = 840 different ways

18) Five candidates must be assigned to the positions of president, vice-president, treasurer, secretary, and receptionist of a club, with one candidate assigned to exactly one position and one position assigned to exactly one candidate.In how many different ways can the positions be filled?

19) Six horses compete in a horse race.If there are no ties and all six horses finish the race, in how many different ways can the race end?

20) Define FACTORIAL NUMBERS.Examples.

†††††† 0! = 1

†††††† 1! = 1

†††††† 2! = (2)(1) = 2

†††††† 3! = (3)(2)(1) = 6

4! = (4)(3)(2)(1) = 24

5! = (5)(4)(3)(2)(1) = 120

6! = (6)(5)(4)(3)(2)(1) = 720

7! = (7)(6)(5)(4)(3)(2)(1) = 5040

8! = (8)(7)(6)(5)(4)(3)(2)(1) = 40,320

9! = (9)(8)(7)(6)(5)(4)(3)(2)(1) = 362,880

10! = (10)(9)(8)(7)(6)(5)(4)(3)(2)(1) = 3,628,800

11! = (11)(10)(9)(8)(7)(6)(5)(4)(3)(2)(1) = 3,991,680

12! = (12)(11)(10)(9)(8)(7)(6)(5)(4)(3)(2)(1) = 47,900,160

 

21) Interpret factorial numbers using elements of sets.

22) a) A voting system consists of candidates, {A, B}.In how many ways can voters rank these two candidates?List all the possibilities.

†††††† b) A voting system consists of three candidates, A, B, C.In how many ways can voters rank these three candidates?List all the possibilities.

†††††† c) A voting system consists of four candidates, {A, B, C, D}.In how many ways can voters rank these four candidates?List all the possibilities.

23) Define PERMUTATIONS of elements of a set.

24) How many permutations of candidates A,B,C,D are there?List the possible permutations.

25) Suppose that three candidates {Amal, Marie, Sy} are available to fill in the positions of President, Vice-president, and Secretary of a particular club.If exactly one candidate must be assigned to each position and exactly one position to each candidate, list all possible permutations of the three candidates.

26) Assuming no ties, in how many different ways can a swimming race with of eight contestants end?

27) In how many different ways can five people lineup to get on a bus (one at a time)?

28) Define the number of permutations of the n elements of a set taken k at a time.

29) Suppose that there are ten candidates for three offices: President, Vice-president, and Secretary with exactly one person in each office and exactly one office for a person.In how many different ways can the three offices be filled?

30) A particular kind of competition awards $200,000 to the first place, $100,000 to the second place, and $50,000 to the third place.If there are twelve competitors, in how many different ways can the three prizes be awarded?

31)Evaluate:a) †††††††b) †††††††c) ††††††††d)

32) In how many ways can the eight members of a committee select a chairman and a vice-chairman?

33)Define COMBINATIONS.

34) a) Compute the number of combinations of all the n elements of a finite set.

††††† b) Compute the number of combinations of all n elements of a finite set taken k at a time (k £ n).

35) How many different subsets of two elements can be formed from {A, B, C, D}?List the 2-element subsets.

36) How many different 3-element subsets can be formed from the set {A, B, C, D, E}?List the 3-element subsets.

37) Define DUALITY :††

38) How many 3-member committees can be formed from a group of 10 individuals?

39) At a local hospital, there are seven open nursing jobs. If 30 registered nurses apply for the jobs, in how many ways can 7 nurses be selected from the 30 applicants?

40) In the Florida Lotto game, in how many ways can a player match three of the six winning numbers?

41) How many committees of four women and two men can be formed from a group of 35 individuals of which ten are men and the rest are women?

42) Consider a group of 8 Democrats, 6 Republicans, and 5 Independents.

††††† a) How many three-member committees can be formed if each party must be represented?

††††† b) How many six-member committees can be formed if two members of each party must be in the committee?

43) a) An election uses sequential pairwise voting and there are 4 candidates: A, B, C, D.How many different agendas are possible?

b) An election uses sequential pairwise voting and there are 10 alternatives, how many different agendas are possible?

44) a) In an election that uses the Condorcetís method, we must examine all possible one-on-one contests between pairs of candidates.If there are three

†††††††††† candidates, how many such contests are there?

†††††† b) If there are twelve candidates, how many such contests are there?

 

45) A group of six girls and five boys must sit in a straight row.

††††††† a) In how many ways can the eleven boys and girls sit in a straight row?

††††††† b) If the six girls must sit together, in how many ways can the eleven boys and girls sit in a straight row?

††††††† c) If the six girls must sit together and the five boys must sit together too, in how many ways can the eleven boys and girls sit in a straight row?

 

 

EXERCISES

COUNTING PRINCIPLES

[1] Find the number of subsets of the set {A, B, C}

[2] Find the number of subsets of the set {A, B, C, D, E}

[3] Find the number of subsets of the set {A, B, C, D, E, F, G, H, I, J}

[4] List all possible subsets of the set {A, B, C}

[5] List all possible subsets of the set {A, B, C, D}

[6] List all possible subsets of the set {A, B, C, D, E}

[7] A particular code consists of one digit followed by one letter from the Alphabet.How many different codes of this kind can be formed?

[8] A woman has 12 different skirts and 16 different blouses.How many different outfits can she wear?

[9] A restaurant offers nine appetizers and twelve main courses.In how many ways can a customer order a meal consisting of one appetizer and one main course?

[10] A man has 5 shirts, 4 ties, 6 pairs of pants, 8 pairs of socks, and 3 pairs of shoes.Find the number of different outfits that he can form if an outfit includes one item from each group of items.

[11] A woman has 5 blouses, 8 skirts, 20 pairs of shoes, and 10 purses.If an outfit includes exactly one item from each group of items, how many different outfits can she wear?

[12] How many four-digit numbers can be formed from the set of ten digits {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} if the first digit cannot be zero and digits cannot repeat.

[13] A bank security system requires customers to choose a five-digit PIN (personal identification number).If all digits must be different (no repetitions allowed) and nonzero, how many different five-digit PINs can be formed?

[14] How many (three-digit) area codes exist if the first digit cannot be zero or one, the second digit must be zero or one, (no restriction on the third digit)?

[15] A true-false exam consists of six questions.If an unprepared student takes this exam, in how many ways can the student answer the six questions?

[16] A multiple-choice exam consists of six questions, each question having options a, b, c, d, with only one option being correct.If an unprepared student takes this exam, in how many ways can the student answer the six questions?

[17] Six performers are to present their comedy acts on a weekend evening at a comedy club.In how many different ways can they schedule their appearances?

[18] In how many different ways can nine books be arranged in a shelf?

[19] Ten singers are to perform on a weekend evening at a night club.How many different ways are there to schedule their appearances?

[20] In how many different ways can a police department arrange eight suspects in a police lineup if each lineup contains all eight people?

[21] Four candidates are to be assigned to the positions of President, Vice-president, Treasurer, and Secretary.Each office must be assumed by exactly one person and each person must be assigned to exactly one office.In how many different ways can this be done?

[22] There are three routes from College Park to Baltimore and five routes from Baltimore to New York.Find how many routes there are from College Park to New York.

[23] How many different outfits consisting of a coat and a hat can be chosen from sixcoats and four hats?

[24] How many different license plates consisting of three letters followed by three digits are possible?

[25] The Spoiled & Rotten Restaurant offers 8 choices of entrees, five choices of salads, ten choices of beverages, and four choices of desserts.How many different meals can a customer order?

[26] a)In how many ways can six people be arranged in a line to take a picture?

†††††† b)In how many ways can they be arranged if three of the six insist on appearing

†††††††††††† one next to the other?

†††††† c)In how many ways can they be arranged if two of the six refuse to appear one

†††††††††††† next to the other?

[27] In how many ways can five books be arranged on a bookshelf?

[28] How many rearrangements of the letters L, S, A, B are possible?†† List allpossible rearrangements.

[29] Twelve participants enter an Olympic event.In how many ways can they be awarded the Gold Medal to the first place, the Silver Medal to the second place, and the Bronze Medal to the third place?

[30] In how many ways can a 24-member team select a captain and an assistant captain?

[31] A group of 12 investors wants to elect a president, a vice-president, a secretary, and a treasurer from the twelve members.In how many ways can this be done?

[32] In a particular state, one type of license plate consists of three uppercase letters followed by three digits.(a)If repetitions are allowed, how many license plates can be formed?(b)If repetitions are not allowed, how many license plates can be formed?

[33] A restaurant offers four soups, twelve entrees, nine beverages, and nine desserts.In how many ways can a customer order a meal if three of the desserts are pies and customers never order pies?

[34] A group of 12 investors wants to form a needs a four-person committee.Howmany four-person committees can be formed from the 12 members?

[35] A group of 12 people consists of five Republicans and the rest are Democrats.A five-person committee must be formed from these 12.

a) How many five-person committees can be formed?

b) How many of these committees consist of two Republicans?

c) How many of these committees consist of at least two Republicans?

d) How many of these committees consist of at most two Republicans?

[36] The Florida Lotto Game consists of choosing six numbers in any order from the set {1, 2, 3, Ö, 53}.Any player who matches the six winning numbers in any order wins the jackpot.

a) How many different choices of six numbers exist?

b) In how many ways can a player match three of the six winning numbers?

[37] A group of five girls and three boys must sit on a straight row.In how many ways can they sit if: A) no restrictions are imposed? B) the three boys must sit together one next to the other? C) the girls must sit together (one next to the other) and the boys too.

[38] A group of nine faculty members consists of 3 members from the English department, 4 from Business, and the rest from Mathematics.A three-member committee must be formed from the group of 9.Find the number of three-member committees if:

†††† a) no restrictions are imposed

†††† b) each department must have a representative

c) no restrictions are imposed except that one member must be the chairperson, another member must be the vice-chair, and the third must be the note-taker.

[39] Six performers are to present their comedy acts on a weekend evening at a comedy club. How many different ways are there to schedule their appearances?

[40] A pizza can be ordered with three choices of size (small, medium, or large), four choices of crust (thin, thick, crispy, or regular), and six choices of toppings (ground beef, sausage, pepperoni, bacon, mushrooms, or onions).How many one-topping pizzas can be ordered?

[41] A medical researcher needs 6 people to test the effectiveness of an experiment drug.If 16 people have volunteered for the test, in how many ways can 6 people be selected?

[42] Suppose you are taking a multiple-choice test that has six questions.Each of the questions has four answer options, with exactly one correct answer per question.If you select exactly one of these four choices for each question and leave nothing blank, in how many ways can you answer the questions?

[43] If three-digit codes are formed from the set {0, 3, 4, 5, 6, 7},

a)     how many codes can be formed if repetitions are allowed?

b)    how many codes can be formed if no repetitions are allowed?

c)     how codes are three-digit odd numbers?

d)    how many codes are three-digit numbers greater than 414?

[44] In how many different ways can a police department arrange eight suspects in a police lineup if each lineup contains all eight people?

[45] A club with ten members is to choose four officers Ė president, vice-president, secretary, and treasurer.If each office is to be held by exactly one person and no person can hold more than one office, in how many ways can those offices be filled?

[46] In how many ways can a committee of five women and four men be formed from a group of 18 people in which ten are women and the rest are men?

[47] In a medical study patients are classified by blood type and blood pressure.If the blood types include {AB+, AB-, A+, A-, B+, B-, O+, O-} and the blood pressures could be {normal, low, high}, find the number of ways in which a patient can be classified.

[48] a) How many distinct permutations can be made from the letters of the word exam?

†††††† b)How many of these permutations end with the letter x?

[49] a)A voter has to rank 4 alternatives.How many different preference list ballots can s/he make?

†††††† b)If there are seven candidates in an election, how many different preference list ballots can each voter make?

[50] Simplify each of the following:

†††††† a)8!††††††††††††† b) c)†††††† †††††d)†††††† ††††e)

[51] How many permutations are there of the letters ABCDE?

[52]Evaluate:

††††††††† a)††††††††††††††††††††††††††††††† b)††††††††††††††††† c)††††††††††††††††††††††††††††† d)

 

[53] Prove that

[54] a) How many subsets does {a, b, c, d, e} have?

††††††† b) How many subsets of {a, b, c, d, e} contain exactly three elements?

[55] a) How many subsets does {a, b, c, d} have?

†††††† b) How many subsets of {a, b, c, d} contain exactly two elements?

[56] How many subsets of size 4 does a set with cardinality 10 have?

[57] a) From a committee of 8 members, 3 must be chosen to form a subcommittee.In how many different ways can this be done?

 ††††††† b) From a committee of 8 members, 3 must be chosen for the positions of chair, vice-chair, and recording secretary.In how many different ways can this be done?

[58] a) If an election uses the sequential pairwise voting method and there are 4 alternatives: A, B, C, and D, how many different agendas are possible?

†††††† b) If an election uses the sequential pairwise voting method and there are 6 alternatives: A, B, C, D, E, and F, how many different agendas are possible?

†††††† c) If an election uses the sequential pairwise voting method and the alternatives areA, B, C, D, and E, how many different agendas are possible?

[59] a) In an election that uses the Condorcetís method, we must examine all possible one-on-one contests between pairs of candidates.If there are four candidates, how many such contests are there?

††††††† b) In an election that uses the Condorcetís method, we must examine all possible one-on-one contests between pairs of candidates.If there are six candidates, how many such contests are there?

††††††† c) In an election that uses the Condorcetís method, we must examine all possible one-on-one contests between pairs of candidates.If the candidates are A, B, C, D, and E, how many such contests are there?

 

 

ANSWERS

[1]††=8†† subsets;†††† [2]††=32subsets;†††† [3]††=1024subsets

[4]f, {A}, {B}, {C}, {A, B}, {A, C}, {B, C}, {A, B, C}

[5]f, {A}, {B}, {C}, {D}, {A, B}, {A, C}, {A, D}, {B, C}, {B, D},

††††††† {C, D}, {A, B, C}, {A, B, D}, {A, C, D}, {B, C, D}, {A, B, C, D}

[6]f, {A}, {B}, {C}, {D}, {E}, {A, B}, {A, C}, {A, D}, {A, E}, {B, C},

†††††† {B, D}, {B, E}, {C, D}, {C, E}, {D, E}, {A, B, C}, {A, B, D},

†††††† {A, B, E}, {A, C, D}, {A, C, E}, {A, D, E}, {B, C, D}, {B, C, E},

†††††† {B, D, E}, {C, D, E}, {A, B, C, D}, {A, B, C, E}, {A, B, D, E},

†††††† {A, C, D, E}, {B, C, D, E}, {A, B, C, D, E,}

[7] (10)(26) = 260 codes;†† [8] (12)(16) = 192 outfits;†† [9] (9)(12) = 108 meals

[10](5)(4)(6)(8)(3) = 2880 outfits;†††† [11](5)(8)(20)(10) = 8000 outfits;

[12] (9)(9)(8)(7) = 4536 4-digit numbers;[13] (9)(8)(7)(6)(5) = 15120 PINs;

[14](8)(2)(10) = 160 area codes;[15]= 64 ways;[16]= 4096 ways

[17]6! = 720 ways;†† [18]9! = 362,880 ways;††† [19]10! = 3,628,800 ways

[20]8! = 40,320 ways;††† [21]4! = 24 ways;††† [22](3)(5) = 15 routes;

[23](6)(4) = 24 outfits;†† [24](26)(26)(26)(10)(10)(10) = 17,576,000 plates

[25](8)(5)(10)(4) = 1600 meals;†† [26]a)6! = 720 ways;b)(4!)(3!) = 144

†††††††† ways;c)6! Ė (5!)(2!) = 480 ways;††† [27]5! = 120 ways;

[28]4! = 24 ways;LSAB, LSBA, LASB, LABS, LBSA, LBAS,

†††††††† SLAB, SLBA, SALB, SABL, SBLA, SBAL, ALSB, ALBS, ASLB,

†††††††† ASBL, ABLS, ABSL, BLSA, BLAS, BSLA, BSAL, BALS, BASL;

[29]= 1320 ways;††† [30]= 552 ways;††† [31]= 11,880 ways;

[32]a)(26)(26)(26)(10)(10)(10) = 17,576,000 plates;

††††††† b)(26)(25)(24)(10)(9)(8)=11,232,000 plates;††

[33](4)(12)(9)(6) = 2592 meals;††† [34]= 495 committees;†††

[35]a)= 792 five-person committees;b)= 350 committees;

†††††††† c)= 596 committees;

†††††††† d)= 546 committees;

[36]a)= 22,957,480 possibilities;b)= 324,300 ways

[37]a)8! = 40,320 ways;b)(6!)(3!) = 4,320 ways;c)(2)(5!)(3!) = 1440 ways

[38]a)= 84 ways;†† b)(3)(4)(2) = 24 ways;†† c)= 504 ways

[39]6!=720 ways;†† [40](3)(4)(6) = 72 one-topping pizzas;†† [41]= 8008

[42]= 4096 ways;†† [43]a)(6)(6)(6) = 216 codes;†† b)(6)(5)(4) = 120 codes;

†††††††† c)(6)(6)(3) = 108;d)(1)(5)(6) + (3)(6)(6) = 138;††† [44]8! = 40,320 ways;

[45]= 5040 ways;†† [46]= 17,640 ways;†† [47](8)(3) = 24 ways;

[48]a)4! = 24;†† b)(3)(2)(1)(1) = 6;†† [49]a)4! = 24 ballots;b)7! = 5040 ballots

[50]a)40,320;b)43,680;c)210;d)24,024;e)495;†† [51]5! = 120 permutations

[52]a)5040;b)210;c)30;d)15;†† [53]

[54]a)= 32 subsets;b)= 10;†† [55]a)= 16 subsets;†† b)= 6

[56]= 210 subsets of size 4;[57]a)= 56 ways;b)= 336 ways;††

[58]a)4! = 24 agendas;b)6! = 720 agendas;c)5! = 120 agendas;

[59]a)= 6 contests;b)= 15 contests;c)= 10 contests