Maîtrisez l'analyse combinatoire : Permutations, Combinaisons et Arrangements

Question 1 sur 10 10:00

[{"id":70462,"question":"Combien de permutations peut-on former avec les lettres du mot 'MATH' ?","option_a":"12","option_b":"24","option_c":"4","option_d":"6","option_e":"","option_f":"","bonne_reponse":"B","explication":"Le nombre de permutations de 4 lettres distinctes est 4! = 4 × 3 × 2 × 1 = 24.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":70463,"question":"Le nombre de combinaisons de 3 éléments parmi 5 est égal à 10.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"Le nombre de combinaisons est C(5,3) = 5! \/ (3! × 2!) = 10. L'affirmation est donc vraie.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":70464,"question":"Quel est le nombre d'arrangements de 2 éléments parmi 4 ?","option_a":"6","option_b":"12","option_c":"8","option_d":"4","option_e":"","option_f":"","bonne_reponse":"C","explication":"Le nombre d'arrangements A(4,2) = 4! \/ (4-2)! = 4 × 3 = 12.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":70465,"question":"La formule des combinaisons est-elle symétrique ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"Oui, car C(n,k) = C(n, n-k). Par exemple, C(5,2) = C(5,3) = 10.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":70466,"question":"Combien de codes à 4 chiffres peut-on former avec les chiffres de 0 à 9 (répétition autorisée) ?","option_a":"10 000","option_b":"40","option_c":"24","option_d":"9 999","option_e":"","option_f":"","bonne_reponse":"A","explication":"Chaque chiffre a 10 possibilités, donc 10^4 = 10 000 codes possibles.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":70467,"question":"Le nombre de permutations de n éléments distincts est toujours égal à n^n.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"B","explication":"Faux. Le nombre de permutations est n! (factorielle de n), pas n^n.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":70468,"question":"Quelle est la valeur de C(6,2) ?","option_a":"30","option_b":"15","option_c":"20","option_d":"12","option_e":"","option_f":"","bonne_reponse":"B","explication":"C(6,2) = 6! \/ (2! × 4!) = (6 × 5) \/ (2 × 1) = 15.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":70469,"question":"Dans un groupe de 8 personnes, combien de façons peut-on choisir un président et un secrétaire ?","option_a":"56","option_b":"28","option_c":"7","option_d":"8","option_e":"","option_f":"","bonne_reponse":"A","explication":"C'est un arrangement : A(8,2) = 8 × 7 = 56.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":70470,"question":"La formule des arrangements A(n,p) peut s'écrire n × (n-1) × ... × (n-p+1).","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"Vrai. Par exemple, A(5,3) = 5 × 4 × 3.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":70471,"question":"Combien de mains de 5 cartes peut-on former dans un jeu de 52 cartes ?","option_a":"2 598 960","option_b":"52","option_c":"1 337 845 600","option_d":"2 118 760","option_e":"","option_f":"","bonne_reponse":"A","explication":"C(52,5) = 52! \/ (5! × 47!) = 2 598 960.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0}]