Cov Qauv Pascal hauv Combinatorics
Combinatorics yog ib ceg ntawm kev suav lej uas kawm txog txoj kev uas cov khoom tuaj yeem teeb tsa. Ib qho ntawm cov cuab yeej nthuav thiab muaj txiaj ntsig tshaj plaws hauv daim teb no yog Pascal's Pattern, tseem hu ua Pascal's Triangle. Pascal's Triangle yog ib daim duab peb ceg ntawm cov lej uas tsim raws li qee txoj cai thiab muaj kev siv dav hauv ntau qhov chaw ntawm kev suav lej, suav nrog kev xav txog qhov muaj feem, kev xav txog tus lej, thiab, ntawm chav kawm, combinatorics.
Keeb kwm ntawm Pascal lub Triangle
Lub duab peb ceg Pascal tau muab lub npe rau Blaise Pascal, tus kws lej Fabkis hauv xyoo pua 17th. Txawm li cas los xij, nws tau paub rau cov kws lej Is Nrias teb thiab Suav ua ntej Pascal lub sijhawm. Hauv Is Nrias teb, nws tau raug hu ua "Meru-Prastaara," thiab hauv Suav teb nws tau raug hu ua "Yang Hui Triangle," uas tau muab lub npe rau tus kws lej Suav Yang Hui.
Pascal lub qauv peb sab
Pascal lub duab peb ceg pib nrog 1 ntawm lub apex. Txhua kab tom ntej yog tsim los ntawm kev ntxiv ob tus lej hauv kab tam sim ntawd saum nws. Kab thawj zaug muaj tsuas yog ib tus lej, 1. Kab thib ob muaj ob tus lej uas kuj yog 1. Kab thib peb muaj 1s ntawm txhua qhov kawg nrog 2 ntawm lawv, qhov tshwm sim ntawm kev ntxiv ob 1s los ntawm kab dhau los.
Feem ntau, kab nth hauv Pascal's Triangle tuaj yeem sau ua:
1, (n-1)C1, (n-1)C2, …, (n-1)C(n-1), 1
Ntawm no, "kC(n)" yog ib lub cim ua ke uas nyeem ua "n xaiv k" lossis "n xaiv k", uas yog ib daim ntawv sib xyaw ua ke hauv kev lej thiab feem ntau siv rau hauv kev xav txog qhov muaj feem yuav tshwm sim thiab linear algebra.
Cov ntawv thov hauv Combinatorics
1. Kev sib xyaw ua ke
Ib qho ntawm cov ntawv thov tseem ceeb ntawm Pascal's Triangle hauv kev sib xyaw ua ke yog kev suav cov kev sib xyaw ua ke. Kev sib xyaw ua ke yog ib txoj hauv kev xaiv cov khoom los ntawm ib pawg uas tsis suav nrog kev txiav txim. Hauv cov ntsiab lus ntawm Pascal's Triangle, cov nqi hauv kab nth thiab kab kth sawv cev rau n-1Ck-1 kev sib xyaw ua ke.
Piv txwv li, los xam qhov sib xyaw ua ke 5C2 (xaiv 2 ntawm 5), peb tuaj yeem saib kab thib 6 thiab kab thib 3 hauv Pascal's Triangle, uas muab tus nqi 10. Hauv lwm lo lus, muaj 10 txoj hauv kev los xaiv 2 yam khoom los ntawm ib pawg ntawm 5 yam khoom.
2. Kev Hloov Pauv thiab Cov Coefficients Binomial
Pascal's Triangle kuj muaj feem cuam tshuam nrog cov coefficients binomial uas tshwm sim hauv binomial expansion ntawm (x + y)^n. Cov coefficients no yog cov lej uas peb pom hauv Pascal's Triangle. Piv txwv li, qhov kev nthuav dav ntawm (x + y)^3 yog:
(x + y)^3 = 1 x^3 + 3 x^2 y + 3 xy^2 + 1 y^3
Ntawm no, cov coefficients 1, 3, 3, thiab 1 yog cov nqi ntawm Pascal's Triangle hauv kab 4th.
3. Kev Ua Si Feem Puas
Hauv kev tshawb fawb txog qhov muaj feem yuav tshwm sim, Pascal's Triangle feem ntau siv los txiav txim siab qhov muaj feem yuav tshwm sim ntawm ntau yam txiaj ntsig. Piv txwv li, thaum pov npib plaub zaug, peb xav paub qhov muaj feem yuav tshwm sim ob lub taub hau. Siv Pascal's Triangle, peb tuaj yeem nrhiav tus lej ntawm cov kev sib xyaw ua ke, uas nyob hauv kab thib tsib thiab kem thib peb, uas muab tus nqi rau peb ntawm 6. Yog li ntawd, muaj rau txoj hauv kev kom tau txais ob lub taub hau hauv plaub qhov pov npib.
Cov Khoom Tshwj Xeeb ntawm Pascal's Triangle
Pascal lub duab peb sab kuj muaj ntau yam khoom nthuav thiab xav tsis thoob:
1. Kev Sib Npaug
Daim duab peb ceg ntawm Pascal qhia txog kev sib npaug ntawm cov lej. Kab nth ntawm daim duab peb ceg ntawm Pascal yog sib npaug, yog li nCr = nC(nr).
2. Kev Sib Raug Zoo ntawm Fibonacci
Pascal's Triangle kuj siv tau los txuas cov kab Fibonacci. Cov lej Fibonacci nrhiav tau los ntawm kev ntxiv cov lej ntawm cov kab diagonal uas sib tshuam ntau kab hauv Pascal's Triangle.
3. Kev sib npaug
Pascal's Triangle qhia txog cov qauv sib npaug zoo kawg nkaus. Yog tias peb pleev xim rau cov lej khib thiab cov lej txawm hauv Pascal's Triangle txawv, cov qauv pom zoo nkauj yuav tshwm sim, feem ntau tsim cov fractals.
Kev siv cov qauv Pascal hauv kev sau programming
Pascal's Triangle kuj siv ntau zaus hauv cov algorithms thiab programming. Piv txwv li, peb tuaj yeem tsim Pascal's Triangle siv cov lus programming zoo li Python nrog cov lej hauv qab no:
"" python
def generate_pascals_triangle(n):
daim duab peb ceg = [[1]]
rau kuv nyob rau hauv ntau yam (1, n):
kab = [1]
rau j nyob rau hauv ntau yam (1, i):
kab.append(peb sab[i-1][j-1] + peb sab[i-1][j])
kab.append(1)
daim duab peb ceg.append(kab)
daim duab peb sab rov qab
n = 5
daim duab peb ceg = generate_pascals_triangle(n)
rau kab hauv daim duab peb sab:
luam tawm (kab)
“
Cov lej saum toj no yuav tso tawm thawj kab mus rau kab thib tsib ntawm Pascal's Triangle, uas tuaj yeem siv rau ntau yam kev sib xyaw ua ke thiab kev siv tshuaj xyuas qhov muaj feem.
Xaus
Pascal's Triangle, lossis Pascal's Pattern, yog ib qho cuab yeej muaj zog thiab siv tau ntau yam hauv kev sib xyaw ua ke. Txij li kev suav cov kev sib xyaw ua ke thiab qhov tshwm sim hauv kev ua si qhov tshwm sim mus rau kev txhais cov kev nthuav dav binomial thiab txuas ntau yam kev xav txog lej, Pascal's Pattern muab txoj hauv kev zoo thiab nkag siab yooj yim los daws cov teeb meem nyuaj. Nrog nws cov qauv yooj yim tab sis qhov tob ntawm lej, Pascal's Pattern txuas ntxiv mus kawm thiab siv rau ntau yam kev kawm lej thiab lwm yam kev tshawb fawb.
