Chapitres Maths en ECG1
Chapitres Maths en ECG1
Corrigés d’exercices : Probabilités sur un univers fini en ECG1
Résumé de cours Exercices Corrigés
Cours en ligne de Maths en ECG1
Corrigés – Probabilités sur un univers fini
Exercice 1 :
1) Par définition, il y a tirages possibles.
2) a) Comme on veut que toutes les boules aient un numéro inférieur à cela revient à tirer
boules dans l’ensemble
il y a
tirages possibles.
b) Pour obtenir un tirage de boules dont le plus grand numéro est égal à
on choisit
boules parmi les boules numérotées de
à
de
façons et on ajoute la boule
à ces
boules.
Lorsque le nombre de tirages de
boules dont le plus grand élément est égal à
est égal à
![Rendered by QuickLaTeX.com B_k](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-985b142d3b1faa31c8f477dce5fecaf6_l3.png)
![Rendered by QuickLaTeX.com p](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-3bf85f1087e9fbed3a319341134ac1a2_l3.png)
![Rendered by QuickLaTeX.com k](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-3422b6bb5c160593658b7c39425d9880_l3.png)
![Rendered by QuickLaTeX.com A_k](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-997dd0945420c88c8e3c38e3fd916bd1_l3.png)
![Rendered by QuickLaTeX.com p](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-3bf85f1087e9fbed3a319341134ac1a2_l3.png)
![Rendered by QuickLaTeX.com k,](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-2b7a60019863c02bac48577a6b8827e5_l3.png)
![Rendered by QuickLaTeX.com B_k = A_k \cup B_{k - 1}.](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-f4a49c6f9d3d15b3ecae817397ef017a_l3.png)
![Rendered by QuickLaTeX.com \mathrm{card} \left( B_k \right) = \mathrm{card} \left( A_k \right) + \mathrm{card} \left( B_{k - 1} \right),](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-1da0b8e887cc9363ee095cdb8bddcc29_l3.png)
![Rendered by QuickLaTeX.com \mathrm{card} \left( A_k \right) = \displaystyle\binom{k}{p} - \displaystyle\binom{k - 1}{p}](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-06d5c9c37d0393a6e644511b99f7ac99_l3.png)
![Rendered by QuickLaTeX.com \mathrm{card} \left( A_k \right) = \displaystyle\binom{k - 1}{p - 1}.](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-5fc40d6be71596673ffd997e295f2c55_l3.png)
c) Soit l’ensemble des tirages de
boules parmi
boules numérotées de
à
. On rappelle que
est l’ensemble des tirages dont le plus grand numéro vaut
.
![Rendered by QuickLaTeX.com p](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-3bf85f1087e9fbed3a319341134ac1a2_l3.png)
![Rendered by QuickLaTeX.com n \ge k \ge p.](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-b7b3a195abc9311598c99b8b6a7d833d_l3.png)
![Rendered by QuickLaTeX.com \left( A_k \right)_{k \in [\![ p ,n ]\!]}](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-dbdfee6b9d42dfb5fa7567a384243c99_l3.png)
![Rendered by QuickLaTeX.com A.](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-78b76206d2de23b4cdedd624f6f63dde_l3.png)
![Rendered by QuickLaTeX.com \bullet](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-2b6e225d778ccd32cb2bd9cc4eaead9a_l3.png)
![Rendered by QuickLaTeX.com A_k](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-997dd0945420c88c8e3c38e3fd916bd1_l3.png)
![Rendered by QuickLaTeX.com \left( 1 , 2 , \cdots, p - 1, k \right)](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-b38dd72563ce415bc1494aa3bae03022_l3.png)
![Rendered by QuickLaTeX.com \bullet](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-2b6e225d778ccd32cb2bd9cc4eaead9a_l3.png)
![Rendered by QuickLaTeX.com A_k \cap A_{k'} = \emptyset](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-92dbf778d013c80a69400a6a58d37a22_l3.png)
![Rendered by QuickLaTeX.com k \neq k'](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-1631c392fbfc52aa53931271a0eac412_l3.png)
![Rendered by QuickLaTeX.com \bullet](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-2b6e225d778ccd32cb2bd9cc4eaead9a_l3.png)
![Rendered by QuickLaTeX.com \displaystyle\bigcup_{k=p}^n A_k = A.](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-5b73bb270f10ae931371e2e659414941_l3.png)
![Rendered by QuickLaTeX.com \displaystyle\bigcup_{k=p}^n A_k \subset A](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-c747897fa6cb2b11bdfe852ee907d341_l3.png)
![Rendered by QuickLaTeX.com p](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-3bf85f1087e9fbed3a319341134ac1a2_l3.png)
![Rendered by QuickLaTeX.com n,](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-428977d87d8c9ab2c3d60050f95d109b_l3.png)
![Rendered by QuickLaTeX.com j,](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-51223dde5149ec01a8a5c4b09e386482_l3.png)
![Rendered by QuickLaTeX.com p.](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-fa6ef6ec04c2dccd40c7f3e3be899df7_l3.png)
![Rendered by QuickLaTeX.com A_j.](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-6abc954b458c3416546ce8ec4155ac32_l3.png)
Comme (nombres de façons de tirer
boules parmi
) et
(question précédente), on a bien l’égalité souhaitée.
![Rendered by QuickLaTeX.com p](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-3bf85f1087e9fbed3a319341134ac1a2_l3.png)
![Rendered by QuickLaTeX.com n](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-b170995d512c659d8668b4e42e1fef6b_l3.png)
![Rendered by QuickLaTeX.com \dfrac{n!}{\left( n - p \right)!}.](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-38765c3968e9a960073f79af9bd4c462_l3.png)
![Rendered by QuickLaTeX.com p](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-3bf85f1087e9fbed3a319341134ac1a2_l3.png)
![Rendered by QuickLaTeX.com n](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-b170995d512c659d8668b4e42e1fef6b_l3.png)
![Rendered by QuickLaTeX.com 2](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-e584dd0bab4e6c8efc164939c28db757_l3.png)
![Rendered by QuickLaTeX.com p - 1](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-923150ba0b91f78280afbecbf7fc3a20_l3.png)
![Rendered by QuickLaTeX.com n - 1](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-169eb2c045bbf4182414e09e9f545fc3_l3.png)
![Rendered by QuickLaTeX.com \left\{ 1, 3, \cdots , n \right\},](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-940674af7fb26f2f5cabd1760d025605_l3.png)
![Rendered by QuickLaTeX.com \dfrac{\left( n - 1 \right)!}{\left( n - 1 - \left( p - 1 \right) \right) !} = \dfrac{\left( n - 1 \right)!}{\left( n - p \right)!}.](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-14c3260828a1932da4d01b8fe4bfbe4c_l3.png)
![Rendered by QuickLaTeX.com p](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-3bf85f1087e9fbed3a319341134ac1a2_l3.png)
![Rendered by QuickLaTeX.com n](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-b170995d512c659d8668b4e42e1fef6b_l3.png)
![Rendered by QuickLaTeX.com n^p.](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-b8bf2223133234702ffb242d7cad13dd_l3.png)
b) Se donner un tirage avec remise de boules tel que le premier numéro soit strictement inférieur au dernier revient
![Rendered by QuickLaTeX.com \bullet](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-2b6e225d778ccd32cb2bd9cc4eaead9a_l3.png)
![Rendered by QuickLaTeX.com 1 \le i < j \le n,](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-e3d45e69b1a30f780888c7bebd32912e_l3.png)
![Rendered by QuickLaTeX.com \left\{ 1 , \cdots , n \right\}](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-93f89af4f882eadd7da4581d7a27aaf0_l3.png)
![Rendered by QuickLaTeX.com \displaystyle\binom{n}{2} = \dfrac{n \left( n - 1 \right)}{2}](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-a3bb5df18af89ad27b457631b9d32d76_l3.png)
![Rendered by QuickLaTeX.com \bullet](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-2b6e225d778ccd32cb2bd9cc4eaead9a_l3.png)
![Rendered by QuickLaTeX.com p - 2](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-5aa558c53a7f25aa47ab5cf124106b3c_l3.png)
![Rendered by QuickLaTeX.com n](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-b170995d512c659d8668b4e42e1fef6b_l3.png)
![Rendered by QuickLaTeX.com 2](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-e584dd0bab4e6c8efc164939c28db757_l3.png)
![Rendered by QuickLaTeX.com n - 1](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-169eb2c045bbf4182414e09e9f545fc3_l3.png)
![Rendered by QuickLaTeX.com n^{p - 2}](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-1c4a9bdd2440fd69dc2cca452c606616_l3.png)
![Rendered by QuickLaTeX.com \dfrac{n \left( n - 1 \right)} {2} n^{p - 2} = \dfrac{n^{p - 1} \left( n - 1 \right)}{2}.](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-1c3789e9ad2b83c1df5c52b4ac186b94_l3.png)
COURS DE MATHS
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Exercice 2 :
1) a) Il ne pas y avoir de surjection car l’ensemble d’arrivée contient plus d’éléments que l’ensemble de départ. Ainsi si
b) Lorsque on sait que si
est surjective, alors
est injective donc
est bijective.
Or, il y a bijections d’un ensemble à
éléments dans un ensemble à
éléments. Ainsi
Si il n’y a qu’une seule application : l’application qui à chaque élément de l’ensemble de départ associe l’unique élément de l’ensemble d’arrivée. Cette application étant surjective, il y a
application et
c) Rappelons qu’il y a applications d’un ensemble à
éléments vers un ensemble à
éléments, disons
. Parmi toutes ces applications, seules deux ne sont pas surjectives : celle qui à tout élément de l’ensemble de départ associe l’élément
et celle qui à tout élément de l’ensemble de départ associe l’élément
![Rendered by QuickLaTeX.com S_2^p = 2^p - 2.](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-13c2b39de4ae06ed213fd5327e35de3f_l3.png)
d) Définir une surjection d’un ensemble à
éléments vers un ensemble
à
éléments revient :
![Rendered by QuickLaTeX.com \bullet](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-2b6e225d778ccd32cb2bd9cc4eaead9a_l3.png)
![Rendered by QuickLaTeX.com E](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-764e1c770271f92700e1a4fbce46c668_l3.png)
![Rendered by QuickLaTeX.com 2](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-e584dd0bab4e6c8efc164939c28db757_l3.png)
![Rendered by QuickLaTeX.com n + 1](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-d765457d0ddab8a9a397a2ca4bd1f567_l3.png)
![Rendered by QuickLaTeX.com \displaystyle\binom{n + 1}{2} = \dfrac{n \left( n + 1 \right)}{2}.](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-93046ceb4fffc8f104347613c1f7cb9c_l3.png)
![Rendered by QuickLaTeX.com a, b](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-0357ced152d91599aefcf60b48861b74_l3.png)
![Rendered by QuickLaTeX.com \bullet](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-2b6e225d778ccd32cb2bd9cc4eaead9a_l3.png)
![Rendered by QuickLaTeX.com E \backslash \left\{ b \right\}](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-8e09279ba4dc5beca18ed22bca3d6f1a_l3.png)
![Rendered by QuickLaTeX.com F](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-2510519bbe1660dfdffb4195c7287343_l3.png)
![Rendered by QuickLaTeX.com f \left( b \right) = f \left( a \right)](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-81d14849872b4f5a3bb8f7c207c4e7d1_l3.png)
![Rendered by QuickLaTeX.com n!](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-da0ef996f36e1b32a0f26f6e896e1771_l3.png)
![Rendered by QuickLaTeX.com S_{n}^{n + 1} = \dfrac{\left( n + 1 \right) n}{2} \times n! = \dfrac{n}{2} \left( n+ 1 \right)!.](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-0f8c17ac5236f4b51ff42f181cb6c1c5_l3.png)
2) a) La formule
est facile à vérifier en revenant à la définition des coefficients binomiaux.
![Rendered by QuickLaTeX.com \sum_{k=q}^n \left( - 1 \right)^k \displaystyle\binom{n}{k} \displaystyle\binom{k}{q}](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-e200b0ac3c568e1c9c39caf22047933b_l3.png)
![Rendered by QuickLaTeX.com = \sum_{k=q}^n \left( - 1 \right)^k \displaystyle\binom{n}{q} \displaystyle\binom{n - q}{k - q}](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-b1c9f8ff9fb75c4237277503e5c1e660_l3.png)
![Rendered by QuickLaTeX.com = \displaystyle\binom{n}{q} \sum_{k=q}^n \left( - 1 \right)^k \displaystyle\binom{n - q}{k - q}](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-1898269a141c0117efff563e61e4aadc_l3.png)
![Rendered by QuickLaTeX.com \underset{\text{on a posé} \; l = k - q}{=}](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-17bae0ab7e63fe2c19dbd81533053ea6_l3.png)
![Rendered by QuickLaTeX.com \displaystyle\binom{n}{q} \sum_{l=0}^{n - q} \left( - 1 \right)^{l + q} \displaystyle\binom{n - q}{l}](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-4157a8a4441e1f935f02dc6ebeef8d4b_l3.png)
![Rendered by QuickLaTeX.com = \displaystyle\binom{n}{q} \left( - 1 \right)^q \sum_{l=0}^{n - q} \left( - 1 \right)^l \displaystyle\binom{n - q}{l}](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-8c698c34722d69fbcae34daae0884d03_l3.png)
![Rendered by QuickLaTeX.com = \displaystyle\binom{n}{q} \left( - 1 \right)^q \left( 1 - 1 \right)^{n - q}](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-516016178eb89b8ee41fa32603003a76_l3.png)
![Rendered by QuickLaTeX.com = 0](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-78875bc33735907a10d2ce7a75c624e8_l3.png)
![Rendered by QuickLaTeX.com n - q \ge 1.](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-1e64270037569d79aa3076ca85710d06_l3.png)
b) Il est clair que
![Rendered by QuickLaTeX.com \mathrm{card} \left( f \left( E \right) \right) = k \in [\![ 1, n ]\!],](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-5f7a3362ea50659d3cf92364886155bd_l3.png)
![Rendered by QuickLaTeX.com f \in \mathcal A_k](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-72d50b4901260aeaa1347be0bd063665_l3.png)
![Rendered by QuickLaTeX.com f \in \displaystyle\bigcup_{i=1}^n \mathcal A_i.](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-16d87310162cb09edf98819f9d885598_l3.png)
![Rendered by QuickLaTeX.com i \neq j,](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-0edb07335590e748763579d450905877_l3.png)
![Rendered by QuickLaTeX.com \mathcal A_i \cap \mathcal A_j = \emptyset.](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-c57ca3b7a31ec782a4f9eb9031196b38_l3.png)
![Rendered by QuickLaTeX.com f \in \mathcal A_i \cap \mathcal A_j](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-093e3cb7319335b9bfb31d9d242de848_l3.png)
![Rendered by QuickLaTeX.com \mathrm{card} \left( F \left( E \right) \right)](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-95d2bc5f77544f0c15e5c32da1f2fc18_l3.png)
![Rendered by QuickLaTeX.com i](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-695d9d59bd04859c6c99e7feb11daab6_l3.png)
![Rendered by QuickLaTeX.com j.](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-590ccb304b65f560e32c03468874b8d2_l3.png)
![Rendered by QuickLaTeX.com \left( \mathcal A_i \right)_{i \in [\![ 1 , n ]\!]}](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-711c05bb5d08c7d5ff1e771ffcbed700_l3.png)
![Rendered by QuickLaTeX.com F^E](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-98cc0e87464736dc47c7d0f2350d9dfc_l3.png)
Il nous reste à calculer
![Rendered by QuickLaTeX.com f \in \mathcal A_i,](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-d4e0e94db37d859dcf22c4bc60ffd971_l3.png)
![Rendered by QuickLaTeX.com \mathrm{card} \left( f \left( E \right) \right) = i.](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-ccb7191376a3e4f6f8f27f75d4703292_l3.png)
![Rendered by QuickLaTeX.com i](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-695d9d59bd04859c6c99e7feb11daab6_l3.png)
![Rendered by QuickLaTeX.com \displaystyle\binom{n}{i}](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-3fa0dc5f58295c776d6b5b3c8aab8eb6_l3.png)
![Rendered by QuickLaTeX.com p](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-3bf85f1087e9fbed3a319341134ac1a2_l3.png)
![Rendered by QuickLaTeX.com i](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-695d9d59bd04859c6c99e7feb11daab6_l3.png)
![Rendered by QuickLaTeX.com S_i^p](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-54cb34c94fb683c86811295bd9956cef_l3.png)
![Rendered by QuickLaTeX.com \mathrm{card} \left( \mathcal A_i \right) = \displaystyle\binom{n}{i} S_i^p.](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-5c102cb708a904073d6e26c929f143d7_l3.png)
puis
On somme cette relation entre et
puis on intervertit les deux sommes :
.
![Rendered by QuickLaTeX.com \displaystyle\sum_{k=i}^n \left( - 1 \right)^k \displaystyle\binom{n}{k} \displaystyle\binom{k}{i} = 0](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-2ea6fe43a3a764da2602849a6280bf8d_l3.png)
![Rendered by QuickLaTeX.com 0 \le i \le n- 1.](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-37f8c4f093b8eb5a9ea935812cac92d1_l3.png)
![Rendered by QuickLaTeX.com \displaystyle\sum_{k=0}^n \left( - 1 \right)^k \displaystyle\binom{n}{k} k^p](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-41f76b579cb8dc2e25cc44639d484297_l3.png)
![Rendered by QuickLaTeX.com \underset{i =n}{=}](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-5bdc81664b83398ba5167b9edced4a5d_l3.png)
![Rendered by QuickLaTeX.com S_n^p \left( - 1 \right)^n \displaystyle\binom{n}{n} \displaystyle\binom{n}{n}](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-137d3e2645fa6b84e1c63c822fcbf5e0_l3.png)
![Rendered by QuickLaTeX.com = \left( - 1 \right)^n S_n^p](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-4743a9e3ec0a340217f468a892ec956c_l3.png)
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Lors de la correction de vos exercices, pensez à prendre note de vos erreurs et relisez régulièrement les règles ou méthodes qui vous posent problème. Faites de même pour les autres chapitres au programme d’ECG1 en Maths :