Multiplicative principle or fundamental counting principle> If an event can be divided into two stages, where the 1st stage can occur in m different ways and, for each of these m ways, the 2nd stage can occur in n distinct ways, then the number of ways the event can occur is given by m \ n*ExamplesIn how many different ways can a person enter and then leave a hall that has 10 doors?1st STAGE: Enter —————— 10 ways2nd STAGE: Leave ——————— 10 waysEntering and Leaving the hall: 10 x 10 = 100 ways.There are four roads connecting cities A and B, and three roads connecting cities B and C. In how many distinct ways can one go from A to C, passing through B?Making the trip from A to C can be considered an action made up of two successive stages:1st - go from A to B: we have four possibilities.2nd - go from B to C: for each of the previous possibilities, there are three ways to reach C from B.Thus, by the F.C.P., the desired result is 4 x 3 = 12.With the digits 1, 2, 3, 4, 5, and 6, how many three-digit numbers with distinct digits can we form?Forming a three-digit number can be considered an action made up of three successive stages, namely:1st - choosing the hundreds digit: we have six possibilities.2nd - choosing the tens digit: since there can be no repetition of digits, we must have a digit different from the digit chosen for the hundreds place. Thus, there are five possibilities.3rd - choosing the units digit: we must have a digit different from the two previous ones (hundreds and tens). Thus, there are only four possibilities.By the F.C.P., the desired result is 6 x 5 x 4 = 120 numbers.Simple permutation [1]In permutation, the elements that make up the grouping change order, that is, position.We determine the possible number of permutations of the elements of a set with the following expression:\[ P_n = n! \]\[ P_n = n (n-1) (n-2) ... \]As an example, for each letter whose order we change in the alphabet, the next one will have one fewer place to switch, because that path has already been used. That is why factorial is used to explain the number of possible permutations, because for each element used, the number of possibilities decreases by 1.ExamplesFor the election of a representative with 3 candidates (V, C, and F), we have 6 possible outcomes regarding the candidates' positions, that is, 1st, 2nd, and 3rd place. See below the possible outcomes of this election.Outcome 1Outcome 2Outcome 3Outcome 4Outcome 5Outcome 6VCFVFCCVFCFVFCVFVCIn how many distinct ways can we arrange the models Ana, Carla, Maria, Paula, and Silvia for the production of a promotional photo album?\[ P = n! = 5! = 5 · 4 · 3 · 2 · 1 = 120\]In how many distinct ways can we place six men and six women in single file, starting with a man and ending with a woman?When we start the grouping with a man and end with a woman, we will have:- Six men randomly in the first position.- Six women randomly in the last position.- Permuting the rest in the 10 remaining positions in the middle! Thus:\[ P_n = 6 · 10! · 6 = 6 · 10 · 9 · 8 · 7 · 6 · 5 · 4 · 3 · 2 · 1 · 6 = 130,636,800\]Permutation with repetitionIn this permutation, some elements that make up the experimental event are repeated. When this occurs, we must apply the following formula:\[ P_n = \frac{n!}{n_1!· n_2! · n_3! ·...} \]- n! total number of elements in the event- n_1! onward: total number of repeated elementsIn the end, it is a simple permutation divided by the number of times it will repeat.ExamplesHow many anagrams can be formed with the word CASA:The word CASA has: 4 letters (n) and two vowels that repeat (n1).\[ P_{n, n1} = \frac{4!}{2!} = \frac{4 · 3 · 2 · 1}{2 · 1} = \frac{24}{2} = 12 \]CASAACSAASCAASACSCAACSAAAASCAACSCAASSAACSACAACASUsing the name COPACABANA, calculate the number of anagrams formed, disregarding those in which consecutive letter repetitions occur.In the word COPACABANA, we have four letters A and two letters C, and 10 letters in total.\[ P_{n, n1} = \frac{10!}{4! · 2!} = \frac{10 · 9 · 8 · 7 · 6 · 5 · 4 · 3 · 2 · 1}{4 · 3 · 2 · 1 · 2 · 1} = 75600 \]Simple arrangement [2]In a simple arrangement, the location of each element of the set forms different groupings. We must take into account the order of position of the element and its nature. In addition, we must know that changing the elements' positions creates a distinction between the groupings.> In arrangements, order matters!To find the number of possible arrangements in a grouping of p with n elements, we must use the following formula:\[ A_{n,p} = \frac{n!}{(n-p)!} \]- n = elements- p = GroupingsNote that (n - p) represents the elements not of interest for the grouping. For each possible grouping, it will repeat for each possibility of the groups not of interest, which is why we divide by their permutation at the end.ExamplesHow many two-digit numbers can be formed from the set A = {2, 4, 6, 8, 9?}Notice that we have a set of 5 elements, where we must choose 2 at a time. It can be seen that order is important in this case because 21 is different from 12.\[ A_{n,p} = \frac{5!}{(5-2)!} = \frac{5!}{3!} = \frac{5 · 4 · 3 · 2 · 1}{3 · 2 · 1} = 20 \]Flávia, Maria, Gustavo, and Pedro are participating in a competition in which there are prizes for the top three places (1st, 2nd, and 3rd). What are the possible prize outcomes?- Number of competition participants: n = 4- Number of people in each group (award): p = 3\[ A_{n,p} = \frac{4!}{(4-3)!} = \frac{4!}{1!} = \frac{4 · 3 · 2 ·1}{1} = 24 \]Simple combinationIn a simple combination, in a grouping we change only the order of the distinct elements.> In combination, the order of the elements is irrelevant!For this to be done, we can use the formula:\[ C_{n,p} = \frac{n!}{p!(n-p)!} \]- n = elements- p = GroupingsIt is noted that (n - p) represents the elements that are not of interest for the grouping. For each possible grouping, it will be repeated for each possibility of the groups that are not of interest, which is why we divide by their permutation at the end. It is also clear that we must divide by the repetitions that occur in the grouping of interest, since their order is irrelevant, which is why it is also divided by p!, which represents the permutations of that group.ExamplesIn how many different ways can I separate 10 balls of different colors, placing 2 balls in each bag.- n = 10- p = 2- The order in which the balls are placed is not relevant\[ C_{n,p} = \frac{10!}{2!(10-2)!} = \frac{10 · 9 · 8!}{2!8!} = \frac{10 · 9}{2} = 45 \]João is packing his suitcase to travel and will take two pairs of shorts. Knowing that he has five available (blue, black, white, brown, and striped), in how many different ways can he choose the shorts?Notice that we have a set of 5 elements (shorts), from which we must choose 2 at a time, where it does not matter which pair of shorts he chooses first; what matters is which two pairs of shorts will be chosen.It is enough to realize that taking the blue and the black is the same as taking the black and the blue. Therefore, this is a combination:\[ C_{n,p} = \frac{5!}{2!(5-2)!} = \frac{5 · 4 · 3!}{2!3!} = \frac{5 · 4}{2} = 10 \]ExercisesTo go to the club, Júnior wants to wear a T-shirt, a pair of shorts, and a pair of sneakers. Knowing that he has six T-shirts, four pairs of shorts, and three pairs of sneakers, in how many distinct ways can he dress?A travel agency offers airline tickets for the São Paulo – Miami route through two companies: Varig or Vasp. The passenger can also choose between first class, business class, and economy class. In how many ways can a passenger make such a choice?A dinner will consist of three parts: appetizer, main course, and dessert. In how many distinct ways can it be composed, if there are eight appetizer options, five main course options, and four dessert options?A train car has six doors. In how many distinct ways can a passenger enter the train and leave through a door different from the one they entered through?A test consists of ten multiple-choice questions. In how many distinct ways can the test be answered, if each question has five distinct alternatives?With the digits 1, 2, 4, 6, 8, and 9:How many four-digit numbers can we form?How many four-digit numbers with distinct digits can we form?With the digits 2, 3, 4, 5, 6, and 7:How many four-digit numbers with distinct digits begin with 3?How many even four-digit numbers with distinct digits can we form?How many three-digit numbers with distinct digits exist?With the digits 0, 1, 2, 3, 4, 5, and 6, how many odd four-digit numbers can we form?We want to form numbers divisible by 5, composed of four distinct digits. How many possibilities are there using the digits 0, 1, 2, 3, 4, 5, and 6? (Suggestion: analyze two cases: when the number ends in zero and when it ends in 5.)A thief knows that the secret code of a safe is formed by a sequence of three distinct digits. In addition, he knows that the hundreds digit is equal to 4. If, on average, the thief takes 3 minutes to test one possible sequence, what is the maximum time for the thief to open the safe?In a certain city, automobile license plates consist of a sequence of two distinct letters and three digits. How many plates can be made? (Consider an alphabet with 26 letters.)To meet the increase in the number of vehicles, it was decided to add one digit to the car plates. If the rules for making the plates remain the same as in the previous item, what is the new total number of plates?A student is looking for the integer solutions of the equation 2x = a + b. Knowing that a = {1,2,3,4 and 5} and b = {1,2,3,4,5}, in how many ways can the student choose a and b to obtain integer solutions?How many sets of initials can be formed if all people have only one surname and:Exactly one first name?Exactly two first names?Current automobile license plates consist of seven symbols, being three letters from the 26-letter alphabet, followed by four digits.How many distinct plates can we have without the digit zero in the 1st position reserved for digits?In the set of all possible distinct plates, what percentage of those have the first two letters equal?Braille writing for blind people is a system of symbols in which each character is formed by a matrix of six dots, of which at least one stands out in relation to the others. What is the maximum number of distinct characters that can be represented in this writing system?Determine how many three-digit numbers, multiples of 5, have the hundreds digit belonging to {1,2,3,4} and the remaining digits belonging to {0,5,6,7,8,9}.Determine the number of anagrams that can be formed with the letters of the name ALEMANHA. When filling out a lottery card, André chose the following markings: in 4 columns he marked the number 1, in 6 columns he marked the number 2, and in 3 columns he marked the number 3. In how many distinct ways can André mark the cards? In a futsal tournament, a team obtained 8 wins, 5 draws, and 2 losses in the 15 matches played. In how many different ways could these results have occurred?In a test composed of 20 questions involving T or F, in how many different ways will we have twelve T answers and eight F answers?In a company, fifteen employees applied for the positions of financial director and deputy financial director. They will be chosen through the individual vote of the members of the company’s board. Determine in how many different ways this choice can be made.A telephone number is made up of 8 digits. Determine how many telephone numbers we can form with different digits that start with 2 and end with 8.In a prize drawing urn, there are ten balls numbered from 0 to 9. Determine the number of possibilities in a drawing whose prize is formed by a sequence of 6 digits.In a classroom there are 12 female students, one of whom is named Carla, and 8 male students, one of whom is named Luiz. Committees of 5 female students and 4 male students are to be formed. Determine the number of committees in which Carla and Luiz participate simultaneously.A soccer team is composed of 11 players: 1 goalkeeper, 4 defenders, 4 midfielders, and 2 forwards. Considering that the coach has 3 goalkeepers, 8 defenders, 10 midfielders, and 6 forwards available, determine the number of possible ways this team can be formed.A scientific researcher needs to choose three test subjects from a group of eight test subjects. Determine the number of ways he can make the choice.In the game of basketball, each team enters the court with five players. Considering that a team needs at least 12 players to compete in a championship, and that among these, 2 are guaranteed starters, determine the number of teams the coach can form with the remaining players, assuming they can play in any position.Answer Key [3]726160305¹⁰a) 1296; b) 360a) 60; b) 180648882220216min or 3h and 36mina) 650,000; b) 6,500,00013a) 676; b) 17,576a) 158,184,000; b) 3.85%6348672060,060135,135125,97021020,160151,20011,550661,50056120References[1] https://www.infoescola.com/matematica/analise-combinatoria/[2] http://sabermatematica.com.br/a-diferenca-entre-arranjo-e-combinacao.html[3] http://exercicios.brasilescola.uol.com.br/exercicios-matematica/exercicios-sobre-permutacao-com-elementos-repetidos.htm
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