Grade 12 / Combinatorics, enumeration 30 exercises (including 14 corrected)

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ABC1 ABC2 ABC3 ABC4 ABC5 ABC6 ABC1 ABC2 ABC3 ABC4 ABC5 ABC6 ABC 1. Notions about sets E.7514 In a universe Ω , consider the three events A , B and C shown below. For each question, express the hatched part using the events A , B , C , meeting, intersection and complementary symbols. E.7515 In a universe Ω , consider the three events A , B and C shown below. For each question, we consider a part of the universe Ω named M : 1 M = A ∩ C 2 M = C ∩ A ∩ B ∩ C 3 M = A ∪ B ∪ C 4 M = B ∩ C ∩ A 5 M = A ∩ B ∩ C ∪ B ∩ C ∩ A 6 M = A ∩ B ∩ C ∪ B ∩ A ∩ C ∪ C ∩ A ∩ B Represent each of the sets M in the representations below : E.7575 In a universe Ω , consider the three events A , B and C repre-sented below : Which set designates the hatched portion shown above : 1 A ∩ B ∩ C ∪ A ∩ B ∩ C 2 A ∩ B ∩ C ∩ B ∪ C 3 A ∪ B ∪ C ∩ A ∪ B 4 C ∩ B ∪ C https://chingmath.fr chapExoCorrec/7514 sacados/7514 ABC1 ABC2 ABC3 ABC4 ABC5 ABC6 chapExoCorrec/7515 sacados/7515 ABC1 ABC2 ABC3 ABC4 ABC5 ABC6 chapExoCorrec/7575 sacados/7575 ABC
ABABABAB ABABABAB ABC ABC E.7508 In a universe Ω , consider the two events A and B . 1 Represent in the diagrams below the two events M and N defined by: M = A ∩ B ; N = A ∪ B 2 We wish to establish the equality of sets A ∩ B = A ∪ B : a For any elementary event ! , establish the implication: ! ∈ A ∩ B = ⇒ ! ∈ A ∪ B b For any elementary event ! , establish the implication: ! ∈ A ∪ B = ⇒ ! ∈ A ∩ B E.7516 In a universe Ω , consider the two events A and B . 1 Represent in the diagrams below the two events M and N defined by: M = A ∪ B ; N = A ∩ B 2 We wish to establish the equality of sets A ∪ B = A ∩ B : a For any elementary event ! , establish the implication: ! ∈ A ∪ B = ⇒ ! ∈ A ∩ B b For any elementary event ! , establish the implication: ! ∈ A ∩ B = ⇒ ! ∈ A ∪ B E.7517 In a universe Ω , consider the three events A , B and C . 1 In the diagrams below, represent the two events M and N defined by: M = A ∩ B ∪ C ; N = A ∩ B ∪ A ∩ C 2 We wish to establish the equality of sets : A ∩ B ∪ C = A ∩ B ∪ A ∩ C a For any elementary event ! , establish the implication: ! ∈ A ∩ B ∪ C = ⇒ ! ∈ A ∩ B ∪ A ∩ C b For any elementary event ! , establish the implication: ! ∈ A ∩ B ∪ A ∩ C = ⇒ ! ∈ A ∩ B ∪ C E.10408 In a universe Ω , consider the three events A , B and C . 1 In the diagrams below, represent the two events M and N defined by: M = A ∪ B ∩ C ; N = A ∪ B ∩ A ∪ C 2 We wish to establish the equality of sets : A ∪ B ∩ C = A ∪ B ∩ A ∪ C a For any elementary event ! , establish the implication: ! ∈ A ∪ B ∩ C = ⇒ ! ∈ A ∪ B ∩ A ∪ C b For any elementary event ! , establish the implication: ! ∈ A ∪ B ∩ A ∪ C = ⇒ ! ∈ A ∪ B ∩ C 2. Factorial E.8651 Solve the following equations in N : a n ! = 5040 b ( n + 1)! = 720 c n ! = 72 × ( n − 2)! E.8652 Determine the value of the following ex-pressions : a 9! × 12! 8! × 11! b (15!) 2 13! × 14! 3. Factorial and combinatorial E.4103 There are 26 cards où in which the 26 letters of the alphabet are written. Three cards are drawn successively from this deck without replacement. 1 How many three-letter words (with or without meaning) can be composed? Justify that this number is written : 26! 23! https://chingmath.fr chapExoCorrec/7508 sacados/7508 Loi de Morgan ABABABAB chapExoCorrec/7516 sacados/7516 ABABABAB chapExoCorrec/7517 sacados/7517 Loi de Morgan ABC ABC sacados/10408 ABC ABC sacados/8651 sacados/8652 chapExoCorrec/4103 sacados/4103
2 a How many words beginning with the letter B can be created? b Deduce the probability of the event : A 1 : ˇ The word begins with the letter B ı. 3 Determine the probability of the event : A 2 : ˇ The second letter of the word is the letter B ı. 4 What is the probability of the event : C : ˇ The word contains the letter B ı. E.4165 Below are all the 3 -digit combinations of the first five non-zero natural numbers. 1 ; 2 ; 3 ; 1 ; 2 ; 4 ; 1 ; 2 ; 5 ; 1 ; 3 ; 4 1 ; 3 ; 5 ; 1 ; 4 ; 5 ; 2 ; 3 ; 4 ; 2 ; 3 ; 5 2 ; 4 ; 5 ; 3 ; 4 ; 5 An urn contains six balls numbered from 1 to 6 . A game con-sists of simultaneously drawing three balls from this urn. It is assumed that each draw is independent and that all draws are equiprobable. 1 a Write down all possible combinations of two num-bers parni the first five non-zero natural integers. Jus-tify that the number of such combinations is equal to : 5! 2! · (5 − 2)! b Deduce the number of combinations of three of the first six non-zero natural numbers. c Establish the following equality: 6! 3! · (6 − 3)! = 20 2 It is assumed that the balls numbered 1 to 4 are blue and the others are red. a Writing all combinations of 2 numbers among the first four non-zero natural integers, establish that the num-ber of such combinations is equal to : 4! 2! · (4 − 2)! b Deduce the number of combinations achieving the event : E : ˇ two balls are blue and only one is red ı c Determine the probability of the event E . E.7577 An association is made up of 18 members, 7 men and 11 women. Each year, the management commit-tee must be elected. It consists of a president and two vice presidents. The association’s articles of association stipulate : if the president is a man, the two vice presidents must be women ; if the president is a woman, both vice presidents must be men ; How many different management committees can be formed with association members? a 154 b 385 c 616 d 1386 4. Combination E.4181 Determine the value of the following ex-pressions : a 15 13 + 9 6 b 15 7 + 15 8 − 16 8 E.4182 Solve in N , the equation : 8 k =56 E.7828 Using a calculator, determine the value of the following binomial coefficients : a 5 3 a 12 5 a 8 6 a 7 2 E.4172 Reminders: if n and p are two natural integers such that p n then : n p = n ! p !( n − p )! Prove that for any natural number n and any natural number p such that 1 p n , we have : n p = n − 1 p − 1 + n − 1 p 5. Pascal’s triangle E.131 The three parts of the exercise are independent. The results requested will be given as irreducible fractions. In this exercise, random draws are made from the eight cards that make up the four queens and four kings of a deck of cards. Preliminary : Write Pascal’s triangle giving the numbers n p for n less than or equal to 8. 1 First modality: Three of the eight cards are drawn at random simultane-ously. a Determine the number of possible draws. b Determine the number of draws that include three kings. c Determine the probability of making a draw of three cards of the same level, i.e. three kings and three queens. 2 Second modality: we can use a tree https://chingmath.fr chapExoCorrec/4165 sacados/4165 chapExoCorrec/7577 sacados/7577 sacados/4181 sacados/4182 chapExoCorrec/7828 sacados/7828 chapExoCorrec/4172 sacados/4172 sacados/131 Antilles - 2002 - 7 points - obligatoire
a Calculate the probability of the event R 1 ˇ The first card drawn is a roi ı b Knowing that the first card drawn is a king, calculate the probability of still getting a king for the second card. c Determine the probability of getting two kings. d What is the probability of obtaining two figures of the same level, i.e. two kings or two queens? e What is the probability of not getting two figures of the same level? 3 Third modality: One card is drawn and returned to the eight-card deck before the next draw. The draws are independent. a Calculate the probability of getting a heart when you draw one card from the eight chosen. b Four successive draws are made. i D ˙ finish the probability p 1 of getting a heart four times. ii Determine the probability p 2 of getting a heart ex-actly twice. c Using the calculator, give the number of draws needed for the probability of getting only hearts to be less than 10 − 6 E.129 Consider a set X with 5 elements. We represent the set X as follows : X = a ; b ; c ; d ; e Note n m the number of m -element subparts of a n -element set. 1 a Represent all parts of X containing 2 elements such that element a is not part of it. b Justify that the number of 2-element parts of X know-ing that a is not part of it is 4 2 . 2 a Represent all parts of X containing 2 elements such that a belongs to these sub-parts. b Justify that the number of 2-element parts knowing that a belongs to them is 4 1 3 Deduce that : 4 1 + 4 2 = 5 2 4 Justify that for any natural number m and n such that m<n , we have the relation: n − 1 m − 1 + n − 1 m = n m E.5385 1 Reconstruct Pascal’s triangle until n =7 . 2 Using the table in question 1 , give the values of the following binomial coefficients : a 5 3 a 4 0 a 4 2 a 7 5 3 Using the calculator, determine the value of the following binomial coefficients : a 5 3 a 12 5 a 8 6 a 7 2 6. Combinatorial probability E.4254 An urn contains five balls that are indistinguishable to the touch : two green and three red. Two balls are extracted at random from the urn. Note X the random variable equal to the number of green balls in the draw. 1 Check that P ( X =0)= 3 10 then determine the probability law of the random variable X . 2 Calculate the mathematical expectation of the random variable X . 3 Calculate the probability of the following event : A : ˇ the two balls drawn are of the same color ı E.4259 An urn contains 4 black balls and 3 red balls indistinguishable by touch. Two balls are drawn at random simultaneously. 1 Consider the event : A : ˇ the two balls drawn are of the same color ı. Determine the probability of the event A . 2 Consider the event : A : ˇ only one of the two balls drawn is red ı. Determine the probability of the event B . E.4263 For each of the following questions, one or two of the proposed answers are correct. No justifica-tion is required : 1 One card is drawn at random from a deck of 32 cards. The probability of getting neither an ace nor a spade is equal to : a 5 8 b 21 32 c 11 32 d 3 8 2 Two cards are drawn at random simultaneously from a deck of 32 cards. The probability of getting neither an ace nor a spade is equal to : a 105 248 b 21 2 32 2 c 21 2 32 2 d 5 2 8 2 7. Old baccalauréat yearbooks (before 2012) https://chingmath.fr sacados/129 chapExoCorrec/5385 sacados/5385 sacados/4254 sacados/4259 sacados/4263 Extrait d'Antilles-Guyane Juin 2010
E.3119 A cabinet consists of 10 drawers T 1 , T 2 , . . . , T 10 . One person randomly places a ball in drawers and another is responsible for finding the drawer containing the ball using the following strategy : The person opens the drawer T 1 . If the ball is in drawer T 1 , the search is complete, otherwise the person opens drawer T 2 , and so on suite. . . respecting the order of the drawer numbers. Note that with this strategy, the T 10 drawer is never opened. For i integer between 1 and 10 ( 1 i 10 ) , we call B i the event ˇ the ball is in the drawer T i ı. We denote X the random variable equal to the number of drawers that were opened in order to locate the ball with this strategy. 1 Give the set of possible values of X . 2 a Show that, for i integer between 1 and 8 ( 1 i 8 ) , the event [ X = i ] is the event B i . b Justify that the event [ X =9] is the union of the events B 9 and B 10 . c Determine the probability law of X . d Calculate the mathematical expectation of X . E.3174 For each of the 3 questions, only one of the three propositions is correct. The candidate will indicate on the copy the number of the question and the letter corresponding to the chosen answer. No justification is required. A correct answer earns 1 point ; an incorrect answer deducts 0.5 point ; no answer is counted 0 point. If the total is nega-tive, the mark is reduced to zero. A ballot box contains 10 touch-indistinguishable ballots of 3 kinds : 4 are marked ˇ yes ı ; 3 are marked ˇ non ı ; 3 are marked ˇ white ı. In a first game, the player begins by staking 30 euro cents. He then draws a ballot from the ballot box and puts it back in after reading it. If the drawn ballot is marked ˇ yes ı, the player receives 60 euro cents, if it is marked ˇ non ı, he receives nothing. If the ballot drawn is marked ˇ blank ı, he receives 20 euro cents. Question 1 The game is : a favorable au joueur b défavorable to the player c équitable Question 2 The player plays 4 games independently of each other. The probability that he draws at least once a ballot marked ˇ yes ı is equal to : a 216 625 b 544 625 c 2 5 In a second game, the player simultaneously draws two ballots from the ballot box. Question 3 The probability that he obtains a draw of two ballots of different kinds is equal to : a 4 15 b 11 30 c 11 15 https://chingmath.fr sacados/3119 Antilles-Guyane 4 points septembre 1998. sacados/3174
E.3137 An urn contains 4 white balls and 2 black balls, indistinguishable to the touch. 1 Three successive random draws of a ball are made ac-cording to the following procedure : after each draw if the ball drawn is white, it is returned to the urn and if it is black, it is not returned to the urn. The random variable equal to the number of black balls obtained from the three draws is X . A weighted tree may be helpful. a What are the values taken by X ? b Calculate P ( X =0) . c We now propose to determine P ( X =1) . Show that the probability that the only black ball drawn is obtained on the second draw is equal to 8 45 . Noting that the only black ball can be drawn either on the first, second or third draw, calculate P ( X =1) 2 We take the urn in its initial composition : 4 white balls and 2 black balls indistinguishable to the touch. Let n be a natural number greater than or equal to 3. We now perform n successive random draws of a ball from the urn according to the same procedure : after each draw, if the ball drawn is white, we return it to the urn and if it is black, we do not return it to the urn. Let k be an integer between 1 and n . Let N be the event : ˇ the k -th ball drawn is black and all the others are blanchesı . Let A be the event : ˇ we get a white ball in each of the first k − 1 draws and a black ball at k -ième ı. Let B be the event : ˇ a white ball is obtained in each of the last ( n − k ) tirages ı. Calculate P ( A ) , P A ( B ) and P ( N ) . E.3230 Let p A ( B ) be the conditional prob-ability of the event B knowing that the event A has occurred. An urn contains 4 red balls and 2 black balls indistinguishable by touch. 1 Two balls are drawn at random from the urn without de-livery. Note A 0 the event : ˇ no ball is obtained noire ı We note A 1 the event : ˇ we got only one ball noire ı ; We note A 2 the event : ˇ we got two balls noires ı Calculate the probabilities A 0 , A 1 and A 2 . 2 After this first draw, there are therefore 4 balls left in the urn. Two balls are again drawn at random from the urn. Note B 0 the event : ˇ no black ball was obtained from the draw n o 2 ı We note B 1 the event : ˇ we got a sule black ball at the draw n o 2 ı We note B 2 the event : ˇ we got two black balls on the draw n o 2 ı a Calculate p A 0 ( B 0 ) , p A 1 ( B 0 ) and p A 2 ( B 0 ) . b Deduct p ( B 0 ) . c Calculate p ( B 1 ) and p ( B 2 ) . d A single black ball was obtained in this second draw ; What is the probability of having obtained a single black ball in the first? 3 Consider the event R : ˇ it took exactly both draws for both black balls to be extracted from urne ı. Show that : p ( R )= 1 3 . https://chingmath.fr sacados/3137 sacados/3230
E.4162 Robots are located at the center of gravity O of a triangle with vertices S , I , X . Each robot moves in three successive steps as follows : At each step, it passes through one of the three vertices S , I , and X , then moves to point O . the robots are programmed so that, during a step, the probability of passing through vertex S is equal to that of passing through vertex X , and the probability of passing through vertex S is twice that of passing through vertex I ; The different steps are independent of each other. Passages through O are not taken into account. Part A - A single robot A single robot is located at point O . 1 Demonstrate that at each step, the probability that the robot will pass through vertex I is equal to 1 5 . 2 Let E denote the event : ˇ during the three steps, the robot passes successively through the vertices 3 , S , I , and X in that order ı. Prove that the probability of E is equal to 4 125 . 3 We note F the event : ˇ During the three stages, the robot passes exactly through vertices 3 , S , I , and X in any or-der ı. Determine the probability of F . Part B - Multiple robots Robots are located at point O , their movements being inde-pendent of each other. What is the minimum number n of robots required for the probability of the event : ˇ at least one of the robots passes successively through vertices S , I , and X in that order ı be greater than or equal to 0 ; 99 ? E.3191 We roll a tetrahedral die whose four faces bear the numbers 1 , 2 , 3 and 4 . We read the nmobre on the hidden side. For k ∈{ 1 ; 2 ; 3 ; 4 } , note p i the probability of obtaining the number k on the hidden face. The die is unbalanced so that the numbers p 1 , p 2 , p 3 and p 4 in that order, form an arithmetic progression : 1 Knowing that p 4 =0.4 show that : p 1 = 0.1 ; p 2 = 0.2 ; p 3 = 0.3 2 The die is rolled three times in succession. The throws are assumed to be pairwise independent. a What is the probability of obtaining in order the num-bers 1 , 2 , 4 ? b What is the probability of obtaining three distinct numbers arranged in ascending order? 3 The die is thrown 10 times in succession. The throws are assumed to be independent. We denote X the random variable that counts the number of times où the number 4 is obtained. a For 1 i 10 , express as a function of i the probability of the event [ X = i ] . b Calculate the mathematical expectation of X . Inter-pret the result obtained. c Calculate the probability of the event [ X 1] . A value rounded to the thousandth will be given. 4 Let n be a non-zero natural number. We throw n times the die, the throws again being assumed to be indepen-dent two by two. We note U n the probability of obtaining for the first time the number 4 at the n -th throw. a Show that U n is a geometric sequence and is conver-gent. b Calculate S n = n i =1 U i then study the convergence of the sequence S n . c Determine the smallest integer n such that S n > 0.999 . E.4198 The number of scooter accidents at an intersection is observed over a long period. It is then possible to propose the following model : for n scooters cross-ing the intersection during a year ( n being a large unknown number) , we admit that the random variable S n which totals the number of scooter accidents at this intersection during this year follows a binomial distribution ; we estimate that the mathematical expectation of S n noted E ( S n ) is equal to 10 . Let p be the probability of a scooter being involved in an acci-dent at this intersection during the year under consideration. 1 Calculate p , then justify the equality: P ( S n = k ) = n k · 10 n k · 1 − 10 n n − k où k is a natural number such that 0 k n . 2 a Establish equality: ln P ( S n =0) = − 10 × ln 1 − 10 n − 10 n où ln denotes the neperian logarithm function. Deduce that : lim n ↦→ + ∞ P ( S n =0)=e − 10 b Demonstrate that : P ( S n = k +1) = P ( S n = k ) × n − k n − 10 × 10 k + 1 où k is a natural number such that 0 k n − 1 c Show that if : lim n ↦→ + ∞ P ( S n = k ) = e − 10 · 10 k k ! for 0 k n , then we also have : lim n ↦→ + ∞ P ( S n = k +1) = e − 10 · 10 k +1 ( k + 1)! for 0 k +1 n . d Demonstrate using reasoning by recurrence on the nat-ural number k that : lim n ↦→ + ∞ P ( S n = k ) = e − 10 · 10 k k ! où k is a natural number such that 0 k n . 3 It is assumed that the number n is large enough to admit that e − 10 · 10 k k ! is an acceptable approximation of P ( S n = k ) . Use this approximation to calculate to within 10 − 4 the probability that in this year there will be at least three scooter accidents at this intersection. https://chingmath.fr chapExoCorrec/4162 sacados/4162 sacados/3191 chapExoCorrec/4198 sacados/4198
8. Unclassified financial years E.10638 Determine the two natural numbers a and b such that : n n + 1 ! = a n ! − b n + 1 ! https://chingmath.fr sacados/10638