Trinomial kengayish - Trinomial expansion

Yilda matematika, a trinomial kengayish uch atama yig'indisining kuchini kengaytirishdir monomiallar. Kengayish tomonidan berilgan

{ displaystyle (a + b + c) ^ {n} = sum _ { stackrel {i, j, k} {i + j + k = n}} {n i, j, k} ni tanlang, a ^ {i} , b ^ {; ! j} ; ! c ^ {k},}

qayerda $n$ manfiy bo'lmagan butun son bo'lib, yig'indisi salbiy bo'lmagan indekslarning barcha kombinatsiyalari bo'yicha olinadi $men, j,$ va $k$ shu kabi $men + j + k = n$ .^[1] The trinomial koeffitsientlar tomonidan berilgan

{ displaystyle {n ni tanlang i, j, k} = { frac {n!} {i! , j! , k!}} ,}

Ushbu formulaning maxsus holati multinomial formula uchun $m = 3$ . Koeffitsientlarni umumlashtirish bilan aniqlash mumkin Paskal uchburchagi deb nomlangan uchta o'lchamga Paskal piramidasi yoki Paskalning tetraedri.^[2]

Xususiyatlari

Kengaytirilgan trinomial shartlarning soni quyidagicha uchburchak raqam

{ displaystyle t_ {n + 1} = { frac {(n + 2) (n + 1)} {2}},}

qayerda $n$ trinomial ko'tariladigan ko'rsatkichdir.^[3]

Misol

Bilan trinomial kengayishning misoli ${ displaystyle n = 2}$ bu:

${ displaystyle (a + b + c) ^ {2} = a ^ {2} + b ^ {2} + c ^ {2} + 2ab + 2bc + 2ca}$

Shuningdek qarang

Adabiyotlar

^ Koshi, Tomas (2004), Ilovalar bilan alohida matematik, Academic Press, p. 889, ISBN 9780080477343.
^ Xarris, Jon; Xirst, Jeffri L.; Mossinghoff, Maykl (2009), Kombinatorika va grafikalar nazariyasi, Matematikadan bakalavriat matnlari (2-nashr), Springer, p. 146, ISBN 9780387797113.
^ Rosenthal, E. R. (1961), "Trinomial koeffitsientlar uchun Paskal piramidasi", Matematika o'qituvchisi, 54 (5): 336–338.

[1] Koshi, Tomas (2004), Ilovalar bilan alohida matematik, Academic Press, p. 889, ISBN 9780080477343.

[2] Xarris, Jon; Xirst, Jeffri L.; Mossinghoff, Maykl (2009), Kombinatorika va grafikalar nazariyasi, Matematikadan bakalavriat matnlari (2-nashr), Springer, p. 146, ISBN 9780387797113.

[3] Rosenthal, E. R. (1961), "Trinomial koeffitsientlar uchun Paskal piramidasi", Matematika o'qituvchisi, 54 (5): 336–338.

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