Asnful n Nyuṭun

Tga **tanfalit n usnful n Nyuṭun** yat tanfalit tusnakt iskr tt umusnak amqqran  Isḥaq Nyuṭun fad ad yaf usbuɣlu n kraygat taẓḍurt n kra n usnful. Tṭṭaf ismawn yaḍna zun d **tanfalit n usnful** nɣ **tanfalit n Nyuṭun**.

## Mad tettini
Iɣ gan x d y sin ifrdisn n kra n uzbg (s umdya sin imḍanen ilawn nɣ ismlaln, sin igtfuln, sin isiruwn imkkuẓn nna dar illa nafs tiddi, atg.) nna ɣ tlla tasunflt (ad t-igan mas-d xy = yx — s umdya i isiruwn : y iga-tt isiruw n tulut) s ɣikk ad, i kraygat n amḍan ummid agaman, nṭṭaf : 
(x+y)^n=\sum_{k=0}^n {n \choose k} x^{k} y^{n-k}=\sum_{k=0}^n {n \choose k} x^{n-k} y^k,
Ma ɣ imḍann

{n \choose k}=\frac{n!}{k!\,(n-k)!}

(kra n twal ar tt nttara ula C_n^k) gan imuskirn isnfal, « ! » ɛnan sis uskir d x^{0} afrdis-tiggt n uzbg.

S usnfl ɣ tanfalit y s -y, ar nttafa:

(x-y)^n=\left(x+(-y)\right)^n=\sum_{k=0}^n {n \choose k} x^{n-k} (-y)^k

Imdyatn:

:\begin{array}{lclcl}
n=2,& (x + y)^2 &= x^2 + 2xy + y^2,&(x - y)^2 &= x^2 - 2xy + y^2,\\
n=3,& (x + y)^3 &= x^3 + 3x^2y + 3xy^2 + y^3,&(x - y)^3 &= x^3 - 3x^2y + 3xy^2 - y^3,\\
n=4,& (x + y)^4 &= x^4 + 4x^3y + 6x^2y^2 + 4xy^3 + y^4,&&\\
n=7,& (x + y)^7 &= x^7+7x^6y+21x^5y^2+35x^4y^3+35x^3y^4+21x^2y^5+7xy^6+y^7.&&
\end{array}


## Amdya
Ha asnful ad ittuyskar i id n mẓẓiynin:

\begin{align}
(x+y)^0 & = 1, \\[8pt]
(x+y)^1 & = x + y, \\[8pt]
(x+y)^2 & = x^2 + 2xy + y^2, \\[8pt]
(x+y)^3 & = x^3 + 3x^2y + 3xy^2 + y^3, \\[8pt]
(x+y)^4 & = x^4 + 4x^3y + 6x^2y^2 + 4xy^3 + y^4, \\[8pt]
(x+y)^5 & = x^5 + 5x^4y + 10x^3y^2 + 10x^2y^3 + 5xy^4 + y^5, \\[8pt]
(x+y)^6 & = x^6 + 6x^5y + 15x^4y^2 + 20x^3y^3 + 15x^2y^4 + 6xy^5 + y^6, \\[8pt]
(x+y)^7 & = x^7 + 7x^6y + 21x^5y^2 + 35x^4y^3 + 35x^3y^4 + 21x^2y^5 + 7xy^6 + y^7, \\[8pt]
(x+y)^8 & = x^8 + 8x^7y + 28x^6y^2 + 56x^5y^3 + 70x^4y^4 + 56x^3y^5 + 28x^2y^6 + 8xy^7 + y^8.
\end{align}


## Tummla
Nẓḍar ad nsml tanfalit ad s tinawt n ullus.

Tummla nɣ taflalit igan tamakazt uggar ar tswurri s usfki n imuskirn isnfal {n \choose k} mas iga umḍan n tifulin n k ifrdisn ɣ yat tagrumma nna ɣ illa n ifrdisn. Kudnna nsbuɣla tanfalit

:(x+y)^n=(x+y)(x+y)\cdots(x+y)\qquad (n \text{ twal})
ar nttafa yat timrnit n iynfuln s talɣa n x^{j}y^{k} ma ɣ j d k ar mmaln s trtib ad nit uṭṭun n twal nna ɣ nxtar x nɣ y lliɣ tn nkka art nsbuɣlu. Illa darnɣ s bzziz j = n - k, acku kraygat twal ur ar gis ntxtar y, ar ntxtar x. S tɣarast yaḍna, maḥd darɣ {n \choose k} n tɣarasin ur mrwasnin n lixtiyyar n k twal atig n y zɣ n tinfulin (x+y) isggt ddaw as, aynful x^{n-k}y^{k} ixṣṣa ad ibayn ɣ usbuɣlu d umuskir {n \choose k}.

### Tummla s ullus
Ar nsiggil ad nml masd tanfalit ad n (a+b)^n=\sum_{k=0}^n {n \choose k} a^{k} b^{n-k} tṣḥa s tinawt n ullus \forall n \in \mathbb{N}  d \forall \{a,b\} \in \mathbb{A}^2  s ab=ba (maɣ \mathbb{A}  iga uzbg mknna ira igt)

#### Tazwart
Iɣ  n = 0

(a+b)^0 = 1

\textstyle {0 \choose 0} a^{0}b^{0-0} = 1

Tazwart n tanfalit tṣḥa

#### Taɣaldt
Rad nɣal mas tanfalit ad tṣḥa ar twala n n

 (a+b)^{n+1} = (a+b).(a+b)^{n}

 \Rightarrow (a+b).\sum_{k=0}^{n} \textstyle {n \choose k} a^{k}b^{n-k} 

 \Rightarrow \left(\sum_{k=0}^{n} \textstyle {n \choose k} a^{k+1}b^{n-k}\right) + \left(\sum_{k=0}^{n} \textstyle {n \choose k} a^{k}b^{n+1-k}\right) 

Ad nsrs p = k+1

\Rightarrow \left(\sum_{p=1}^{n+1} \textstyle {n \choose p-1} a^{p}b^{n+1-p}\right) + \left(\sum_{k=0}^{n} \textstyle {n \choose k} a^{k}b^{n+1-k}\right)

 \Rightarrow \left(\sum_{k=1}^{n} \left(\textstyle {n \choose k-1} + \textstyle {n \choose k}\right) a^{k}b^{n+1-k}\right) + \textstyle {n \choose 0}a^{0}b^{n+1} + \textstyle {n \choose n}a^{n+1}b^{0}

 \Rightarrow \left(\sum_{k=1}^{n} \textstyle {n+1 \choose k}a^{k}b^{n+1-k}\right) + \textstyle {n+1 \choose 0}a^{0}b^{n+1} + \textstyle {n+1 \choose n+1}a^{n+1}b^{0}

\Rightarrow \left(\sum_{k=0}^{n+1} \textstyle {n+1 \choose k}a^{k}b^{n+1-k}\right)

#### Amsgra
S tifadiwin lli nfka i a d b, tanfalit (a+b)^n = \sum_{k=0}^{n} \textstyle {n \choose k} a^{k}b^{n-k} tṣḥa i \forall n \in \mathbb{N}  .

## Asmata
Nẓḍar ad nals i usbuɣlu s ullus fad ad nml tanfalit n Laybniz i tazllumt tis-n n kra n ufaris.

Taɣarast tamsuddst n tawlawalt ad aɣ yujjan ad nskr asmata n tulut agtful

(X+Y)^n=\sum_{k=0}^n {n \choose k}X^{n-k}Y^k

s

\prod_{i=1}^n(X+Y_i)=\sum_{k=0}^n\sigma_k(Y_1,\ldots,Y_n)X^{n-k},

maɣ id \sigma_{k} nɛna sisn igtfuln ujjuṛn ifrdasn.

Nẓḍar ula ad nskr asmata n tanfalit ad s timrniyin n m irman ismlaln lan yat tuẓḍurt tummidt n 

\left( \sum_{i=1}^m x_i \right)^n = \sum_{\left|\vec k\right|=n}
{n\choose\vec k} \prod_{i=1}^m x_i^{k_i}d s imksanen ur gin ummidn nɣ gan ummidn izdrn.

## Ẓṛ uggar

* Asaḍuf n id kusinus
* Targadda n Cauchy–Schwarz
* Alugaritm agaman
* Asktiɣmr

## Tisaɣulin
 
 

#### Tisaɣulin
 
 
## 


#### Tinidlisn d izdayn
**<small>(en)</small>** J. L. Coolidge, « The Story of the Binomial Theorem », *Amer. Math. Monthly*, vol. 56, no 3,‎ 1949, p. 147-157 (JSTOR 2305028, Ɣr ɣ wanṭirnit)

**<small>(en)</small>** Graham, Ronald; Knuth, Donald; Patashnik, Oren (1994). "(5) Binomial Coefficients". *Concrete Mathematics* (2nd ed.). Addison Wesley. pp. 153–256. ISBN 978-0-201-55802-9. OCLC 17649857.

**<small>(en)</small>** Bag, Amulya Kumar (1966). "Binomial theorem in ancient India". *Indian J. History Sci*. **1** (1): 68–74.

**<small>(en)</small>** Solomentsev, E.D. (2001) [1994], " Newton binomial ", , *Encyclopedia of Mathematics*, EMS Press

## Amawal

#### Tiguriwin

*Tanfalit = formule
*Asnful = binome
*Amusnak = mathematicien
*Asbuɣlu = developpement
*Taẓḍurt = puissance
*Azbg = anneau
*Afrdis = element
*Amḍan = nombre
*Ilaw = reel
*Asmlal = complexe
*Agtful = polynome
*Isiruw = matrice
*Amkkuẓ = carre
*Tasunflt = commutativite
*Tulut = identite
*Ummid = entier
*Agaman = naturel
*Amuskir = coeficient
*Uskir = factorielle
*Afrdis-tiggt = element unite
*Tinawt = énoncé
*Ullus = recurence
*Asfki = definion
*Amakaz = intuitif
*Tafult = partie
*Tagrumma = ensemble
*Timrnit = somme
*Iynfuln = monômes
*Tazllumt = derive
*Tamsuddst = derive
*Awlawal = variant
*Agtful = polynimiale
*Ujjuṛ = symetrique
*Afrdas = elementaire
*Irm = terme
*Amksan = exposant
*Uzdir = negatif


#### Imawaln
Amawal umniḍ DGLAi n Asinag Agldan n Tussna Tamaziɣt, s kraḍ tutlayin (tamaziɣt tatrart tamɣribit - tafransist - taɛrabt) <small></small>

Amawal Tafsut n tusnakt - MCB Alger, Tiwi Uzzu s sin tutlayin (tamaziɣt taqbaylit - tafransist)