Maîtrisez les sommes de coefficients binomiaux — Quiz interactif

Question 1 sur 10 10:00

[{"id":11298,"question":"Quelle est la valeur de la somme Σ (de k=0 à n) C(n,k) ?","option_a":"0","option_b":"1","option_c":"2^n","option_d":"n^2","option_e":"","option_f":"","bonne_reponse":"C","explication":"La somme des coefficients binomiaux pour un n donné est égale à 2^n, car chaque élément peut être choisi ou non dans un ensemble de n éléments.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":11299,"question":"Le binôme de Newton s'applique uniquement aux entiers positifs.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"B","explication":"Le binôme de Newton s'applique à tout couple de nombres réels ou complexes, pas seulement aux entiers positifs.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":11300,"question":"Quelle formule représente le développement de (a + b)^3 ?","option_a":"a^3 + b^3","option_b":"a^3 + 3a^2b + 3ab^2 + b^3","option_c":"a^3 + b^3 + 3ab(a + b)","option_d":"3(a^3 + b^3)","option_e":"","option_f":"","bonne_reponse":"B","explication":"Le développement de (a + b)^3 suit la formule du binôme de Newton : C(3,0)a^3b^0 + C(3,1)a^2b^1 + C(3,2)a^1b^2 + C(3,3)a^0b^3.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":11301,"question":"La somme Σ (de k=0 à n) k*C(n,k) est égale à :","option_a":"n*2^(n-1)","option_b":"n^2","option_c":"2^n","option_d":"n","option_e":"","option_f":"","bonne_reponse":"A","explication":"En dérivant le binôme de Newton (1 + x)^n par rapport à x et en évaluant en x=1, on obtient cette somme égale à n*2^(n-1).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":11302,"question":"L'identité (1 + x)^n = Σ (de k=0 à n) C(n,k) x^k est toujours vraie.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"Cette identité est vraie pour tout réel x et tout entier naturel n, car elle découle directement du binôme de Newton.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":11303,"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":11304,"question":"La somme Σ (de k=0 à n) (-1)^k * C(n,k) est égale à :","option_a":"0","option_b":"1","option_c":"(-1)^n","option_d":"2^n","option_e":"","option_f":"","bonne_reponse":"A","explication":"En remplaçant x par -1 dans le binôme de Newton (1 + x)^n, on obtient cette somme égale à 0 pour n ≥ 1.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":11305,"question":"Le coefficient binomial C(n,k) est toujours supérieur ou égal à C(n,k+1).","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"B","explication":"C(n,k) ≥ C(n,k+1) n'est pas toujours vrai. Par exemple, C(5,2) = 10 \u003E C(5,3) = 10, mais C(5,1) = 5 \u003C C(5,2) = 10.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":11306,"question":"Quelle est la somme Σ (de k=0 à n) C(n,k)^2 ?","option_a":"C(2n,n)","option_b":"2^n","option_c":"n^2","option_d":"n","option_e":"","option_f":"","bonne_reponse":"A","explication":"Cette somme est égale à C(2n,n), car elle compte le nombre de façons de choisir n éléments parmi 2n en combinant deux ensembles de n éléments.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":11307,"question":"L'identité C(n,k) = C(n,n-k) est toujours vraie.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"Cette identité est vraie car choisir k éléments parmi n revient au même que de laisser de côté n-k éléments.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0}]