Nā ʻano hoʻohālikelike o Pascal ma Combinatorics
ʻO Combinatorics kahi lālā o ka makemakika e aʻo ana i nā ʻano e hiki ai ke hoʻonohonoho ʻia nā mea. ʻO kekahi o nā mea hana hoihoi a pono loa i kēia kahua ʻo ia ke ʻano o Pascal, i ʻike ʻia hoʻi ʻo Pascal's Triangle. ʻO Pascal's Triangle kahi huinakolu o nā helu i kūkulu ʻia e like me kekahi mau lula a he nui nā noi ma nā ʻano like ʻole o ka makemakika, me ke kumumanaʻo probabili, ke kumumanaʻo helu, a, ʻoiaʻiʻo hoʻi, combinatorics.
Ke Kumu o ka Triangle o Pascal
Ua kapa ʻia ka huinakolu o Pascal ma muli o Blaise Pascal, he makemakika Farani o ke kenekulia 17. Eia naʻe, ua ʻike ʻia e nā makemakika India a me Kina ma mua loa o ka manawa o Pascal. Ma India, ua kapa ʻia ʻo "Meru-Prastaara," a ma Kina ua kapa ʻia ʻo "Yang Hui Triangle," i kapa ʻia ma hope o ka makemakika Kina ʻo Yang Hui.
ʻO ke ʻano o ka huinakolu o Pascal
Hoʻomaka ka huinakolu o Pascal me ka 1 ma ka piko. Hoʻokumu ʻia kēlā me kēia lālani ma hope ma ka hoʻohui ʻana i nā helu ʻelua ma ka lālani ma luna pono. Loaʻa i ka lālani mua hoʻokahi wale nō helu, ʻo 1. Loaʻa i ka lālani ʻelua ʻelua mau helu he 1 nō hoʻi. Loaʻa i ka lālani ʻekolu nā 1 ma kēlā me kēia wēlau me ka 2 ma waena o lākou, ʻo ia ka hopena o ka hoʻohui ʻana i nā 1 ʻelua mai ka lālani ma mua.
Ma keʻano laulā, hiki ke kākau ʻia ka lālani nth ma ka Huinakolu o Pascal penei:
1, (n-1)C1, (n-1)C2, …, (n-1)C(n-1), 1
Maanei, ʻo "kC(n)" kahi hōʻailona hui i heluhelu ʻia ʻo "n koho k" a i ʻole "n koho k", ʻo ia kahi haʻilula hui i ka makemakika a hoʻohana pinepine ʻia i ke kumumanaʻo probabili a me ka algebra linear.
Nā noi ma Combinatorics
1. Hoʻohuihui
ʻO kekahi o nā hoʻohana nui o ka Triangle o Pascal i ka combinatorics ka helu ʻana i nā hui pū ʻana. ʻO ka hui pū ʻana he ala ia e koho ai i nā mea mai kahi set kahi i noʻonoʻo ʻole ʻia ai ke kauoha. Ma ke ʻano o ka Triangle o Pascal, ʻo nā waiwai ma ka lālani nth a me ke kolamu kth e hōʻike ana i nā hui pū ʻana o n-1Ck-1.
No ka laʻana, no ka helu ʻana i ka hui pū ʻana o 5C2 (ke koho ʻana i 2 mai loko mai o 5), hiki iā kākou ke nānā i ka lālani 6 a me ke kolamu 3 ma ka Triangle o Pascal, kahi e hāʻawi ai i ka waiwai 10. I nā huaʻōlelo ʻē aʻe, aia he 10 mau ala e koho ai i 2 mau mea mai kahi hoʻonohonoho o 5 mau mea.
2. Nā Permutations a me nā Coefficients Binomial
Pili loa nō hoʻi ka Triangle o Pascal i nā coefficients binomial e kū mai ana ma ka hoʻonui binomial o (x + y)^n. ʻO kēia mau coefficients nā helu a mākou e ʻike ai ma ka Triangle o Pascal. No ka laʻana, ʻo ka hoʻonui ʻana o (x + y)^3 penei:
(x + y)^3 = 1 x^3 + 3 x^2 y + 3 xy^2 + 1 y^3
Maanei, ʻo nā coefficients 1, 3, 3, a me 1 nā waiwai o ka Pascal's Triangle ma ka lālani 4.
3. Pāʻani Kūpono
I ke kumumanaʻo probabilidad, hoʻohana pinepine ʻia ka Triangle o Pascal e hoʻoholo ai i ka probabilidad o nā hopena like ʻole. No ka laʻana, i ka hoʻolei ʻana i kahi kālā ʻehā manawa, makemake mākou e ʻike i ka probabilidad o ka loaʻa ʻana o ʻelua mau poʻo. Ma ka hoʻohana ʻana i ka Triangle o Pascal, hiki iā mākou ke loaʻa ka helu o nā hui like, aia ma ka lālani ʻelima a me ke kolamu ʻekolu, kahi e hāʻawi mai ai iā mākou i ka waiwai o 6. No laila, aia ʻeono mau ala e loaʻa ai ʻelua mau poʻo i nā hoʻolei kālā ʻehā.
Nā Waiwai Kūikawā o ka Huinakolu o Pascal
Loaʻa i ka huinakolu o Pascal nā ʻano hoihoi a kupaianaha like ʻole:
1. Hoʻohālikelike
Hōʻike ka Huikolu o Pascal i ke kaulike o nā helu. ʻO ka lālani nth o ka Huikolu o Pascal he kaulike, no laila nCr = nC(nr).
2. Pilina Fibonacci
Hiki ke hoʻohana ʻia ka Triangle o Pascal e hoʻopili i ka moʻo Fibonacci. Hiki ke loaʻa nā helu Fibonacci ma ka hoʻohui ʻana i nā helu ma nā laina diagonal e hui pū ana me kekahi mau laina ma ka Triangle o Pascal.
3. Ka like ʻana
Hōʻike ka Triangle o Pascal i nā ʻano like ʻole. Inā mākou e kala i nā helu ʻē a me nā helu like ʻole ma ka Triangle o Pascal ma ke ʻano ʻokoʻa, puka mai nā ʻano ʻike maka hoihoi, e hana pinepine ana i nā fractals.
Hoʻokō ʻana o nā ʻano Pascal i ka Polokalamu
Hoʻohana pinepine ʻia ka Pascal's Triangle i nā algorithms a me ka papahana. No ka laʻana, hiki iā mākou ke kūkulu i ka Pascal's Triangle me ka hoʻohana ʻana i kahi ʻōlelo papahana e like me Python me ke code aʻe:
“`python
def generate_pascals_triangle(n):
huinakolu = [[1]]
no i ma ka laulā(1, n):
lālani = [1]
no j ma ka laulā(1, i):
lālani.hoʻopili(huinakolu[i-1][j-1] + huinakolu[i-1][j])
lālani.hoʻohui(1)
triangle.hoʻohui(lālani)
huinakolu hoʻihoʻi
n = 5
huinakolu = hana i ka huinakolu_pascals(n)
no ka lālani ma ka huinakolu:
paʻi (lālani)
".
E hoʻopuka ke code ma luna i nā lālani mua a hiki i ka ʻelima o ka Pascal's Triangle, hiki ke hoʻohana ʻia no nā combinatorics like ʻole a me nā noi loiloi probabili.
Ka hopena
ʻO ka Triangle o Pascal, a i ʻole ke ʻano o Pascal, he mea hana ikaika a maʻalahi hoʻi i ka combinatorics. Mai ka helu ʻana i nā hui pū ʻana a me nā probabilities i nā pāʻani probability a hiki i ka wehewehe ʻana i nā hoʻonui binomial a me ka hoʻopili ʻana i nā manaʻo makemakika like ʻole, hāʻawi ʻo Pascal's Pattern i kahi ala kūpono a maʻalahi hoʻi e hoʻoponopono ai i nā pilikia paʻakikī. Me kona ʻano maʻalahi akā me ka hohonu makemakika kupaianaha, ke hoʻomau nei ke aʻo ʻia ʻana a me ka hoʻohana ʻia ʻana o Pascal's Pattern ma nā ʻano like ʻole o ka makemakika a me nā ʻepekema ʻē aʻe.
