Sommes de coefficients binomiaux : Testez vos connaissances !

Question 1 sur 10 10:00

[{"id":46451,"question":"Quelle est la valeur de C(5,2) ?","option_a":"5","option_b":"10","option_c":"15","option_d":"20","option_e":"","option_f":"","bonne_reponse":"B","explication":"C(5,2) = 5! \/ (2! * 3!) = (5×4)\/(2×1) = 10.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":46452,"question":"La somme des coefficients binomiaux C(n,k) pour k allant de 0 à n est égale à :","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"La somme des coefficients binomiaux C(n,k) pour k allant de 0 à n est égale à 2^n, car c'est le nombre de sous-ensembles d'un ensemble à n éléments.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":46453,"question":"Quelle formule permet de développer (a + b)^4 ?","option_a":"a^4 + 4a^3b + 6a^2b^2 + 4ab^3 + b^4","option_b":"a^4 + b^4","option_c":"4(a^4 + b^4)","option_d":"(a + b)^2 * (a + b)^2","option_e":"","option_f":"","bonne_reponse":"A","explication":"En appliquant la formule du binôme de Newton, (a + b)^4 = C(4,0)a^4 + C(4,1)a^3b + C(4,2)a^2b^2 + C(4,3)ab^3 + C(4,4)b^4.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":46454,"question":"Si C(n,3) = 35, quelle est la valeur de n ?","option_a":"5","option_b":"6","option_c":"7","option_d":"8","option_e":"","option_f":"","bonne_reponse":"C","explication":"C(7,3) = 7! \/ (3! * 4!) = (7×6×5)\/(3×2×1) = 35.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":46455,"question":"La relation de Pascal s'écrit :","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"La relation de Pascal s'écrit C(n,k) + C(n,k+1) = C(n+1,k+1).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":46456,"question":"Quelle est la valeur de la somme C(4,0) + C(4,1) + C(4,2) + C(4,3) + C(4,4) ?","option_a":"10","option_b":"16","option_c":"20","option_d":"32","option_e":"","option_f":"","bonne_reponse":"B","explication":"Cette somme est égale à 2^4 = 16, car elle représente le nombre total de sous-ensembles d'un ensemble à 4 éléments.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":46457,"question":"Si C(n,2) = 28, quelle est la valeur de n ?","option_a":"7","option_b":"8","option_c":"9","option_d":"10","option_e":"","option_f":"","bonne_reponse":"B","explication":"C(8,2) = 8! \/ (2! * 6!) = (8×7)\/2 = 28.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":46458,"question":"La somme des coefficients binomiaux C(n,k) pour k pair est égale à la somme pour k impair.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"C'est vrai pour tout n ≥ 1, car les coefficients binomiaux sont symétriques (C(n,k) = C(n,n-k)).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":46459,"question":"Quelle est la valeur de C(6,3) ?","option_a":"15","option_b":"20","option_c":"25","option_d":"30","option_e":"","option_f":"","bonne_reponse":"B","explication":"C(6,3) = 6! \/ (3! * 3!) = (6×5×4)\/(3×2×1) = 20.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":46460,"question":"La formule du binôme de Newton s'applique uniquement aux entiers naturels.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"B","explication":"Faux. La formule du binôme de Newton s'applique à tout couple (a,b) de nombres réels ou complexes, et à tout entier naturel n.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0}]