Asnful N Nyuṭun

Amnni ad icqqa ad t tfhmt ! Amnni ad gis kra n iwalwn nna ur iẓḍaṛ yan ad tn ifhm iɣ ur issin ɣ umawal iẓlin n tutlayt taclḥiyt tatrart. Mad igan afssay ? Fad ad tfhmt mad gis illan tzḍart ad tawst i yixf nnk s umawal lli illan ɣ izddar akkʷ n tasna. Iɣ t ur tufit, tzḍart ad nn taggʷt ɣ imawaln d isgzawaln.

Tṭṭaf ismawn yaḍna zun d tanfalit n usnful nɣ tanfalit n Nyuṭun.

Iɣ gan ${\displaystyle x}$  d ${\displaystyle y}$  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 ${\displaystyle xy=yx}$  — s umdya i isiruwn : ${\displaystyle y}$  iga-tt isiruw n tulut) s ɣikk ad, i kraygat ${\displaystyle n}$  amḍan ummid agaman, nṭṭaf :

${\displaystyle (x+y)^{n}=\sum _{k=0}^{n}{n \choose k}x^{k}y^{n-k}=\sum _{k=0}^{n}{n \choose k}x^{n-k}y^{k}}$ ,

Ma ɣ imḍann

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

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

S usnfl ɣ tanfalit ${\displaystyle y}$  s ${\displaystyle -y}$ , ar nttafa:

${\displaystyle (x-y)^{n}=\left(x+(-y)\right)^{n}=\sum _{k=0}^{n}{n \choose k}x^{n-k}(-y)^{k}}$ 

Imdyatn:

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

Ha asnful ad ittuyskar i id ${\displaystyle n}$  mẓẓiynin:

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

Nẓḍar ad nsml tanfalit ad s tinawt n ullus.

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

${\displaystyle (x+y)^{n}=(x+y)(x+y)\cdots (x+y)\qquad (n{\text{ twal}})}$ 

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

Tummla s ullus

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

Tazwart

Iɣ ${\displaystyle n=0}$ 

${\displaystyle (a+b)^{0}=1}$ 

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

Tazwart n tanfalit tṣḥa

Taɣaldt

Rad nɣal mas tanfalit ad tṣḥa ar twala n ${\displaystyle n}$ 

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

${\displaystyle \Rightarrow (a+b).\sum _{k=0}^{n}\textstyle {n \choose k}a^{k}b^{n-k}}$ 

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

Ad nsrs ${\displaystyle p=k+1}$ 

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

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

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

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

Amsgra

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

Nẓḍar ad nals i usbuɣlu s ullus fad ad nml tanfalit n Laybniz i tazllumt tis-${\displaystyle n}$  n kra n ufaris.

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

${\displaystyle (X+Y)^{n}=\sum _{k=0}^{n}{n \choose k}X^{n-k}Y^{k}}$ 

s

${\displaystyle \prod _{i=1}^{n}(X+Y_{i})=\sum _{k=0}^{n}\sigma _{k}(Y_{1},\ldots ,Y_{n})X^{n-k}}$ ,

maɣ id ${\displaystyle \sigma _{k}}$  nɛna sisn igtfuln ujjuṛn ifrdasn.

Nẓḍar ula ad nskr asmata n tanfalit ad s timrniyin n ${\displaystyle m}$  irman ismlaln lan yat tuẓḍurt tummidt ${\displaystyle n}$ 

$${\displaystyle \left(\sum _{i=1}^{m}x_{i}\right)^{n}=\sum _{\left|{\vec {k}}\right|=n}{n \choose {\vec {k}}}\prod _{i=1}^{m}x_{i}^{k_{i}}}$$

 

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

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Tinidlisn d izdayn

(en) J. L. Coolidge, « The Story of the Binomial Theorem », Amer. Math. Monthly, vol. 56, n^o 3,‎ 1949, p. 147-157 (JSTOR 2305028, Ɣr ɣ wanṭirnit)

(en) 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.

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

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

Tiguriwin

Imawaln

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