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Résumé Fiche de révision math Terminale combinatoire et dénombrement
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Fiche de révision math Terminale combinatoire et dénombrement
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Combinatoire et dénombrementSpécialité Terminale
Principe additif Généralisation
Si A et B sont deux ensembles : On peut généraliser A1 , A2 , . . . , An des ensembles finis
✧ Si A et B sont disjoints : Card(A ∪ B) = Card(A) + deux à deux disjoints :
Card(B) n
X
✧ Si A et B sont quelconques : Card(A ∪ B) = Card(A) + Card(A1 ∪ A2 ∪ . . . ∪ An ) = Card(Ai )
Card(B) − Card(A ∩ B) i=1
Produit cartésien Nombre de parties d’un ensemble
Le produit cartésien de A et B est noté A × B n un entier naturel.
Card(A × B) = Card(A) × Card(B) Card(P(E)) = 2n
k-listes Factorielle
L’ensemble de tous les k-listes de E est l’ensemble : n est un entier non nul.
Ek = E × E × . . . × E
| {z } n! = 1 × 2 × . . . × (n − 1) × n
k fois
k
Card(E k ) = [Card(E)] = nk Par convention 0! = 1.
Arrangements Permutations
Le nombre d’arrangements de k éléments parmi n (k < n) Le nombre de permutations d’un ensemble fini non vide
: à n éléments est n!
n!
Akn = n × (n − 1) × . . . × (n − k + 1) =
(n − k)!
Combinaisons Propriétés
Le nombre de combinaisons de p éléments parmi n : ✧ 06p6n: ! !
! n n
n Ap n! =
= n = p n−p
p p! p!(n − p)!
✧ n > 1 et 1 6 p 6 n − 1 :
! ! !
n+1 n n
= +
p+1 p p+1
Triangle de Pascal Binôme de Newton
a et b deux réels et n un entier naturel non nul :
n
!
n
X n p n−p
(a + b) = a b
p=0
p
Coefficients binomiaux :
n
!
X n
= 2n
p=0
p
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