Mienzaniso yeMibvunzo yeKukurukura Mitemo yeKuzadza Nzvimbo
Mutemo wekuzadza nzvimbo, kana mutemo wekuisa, ipfungwa huru mumasvomhu uye mukana unobatsira zvikuru mumamiriro ezvinhu akawanda. Mutemo uyu unowanzo shandiswa pakuronga zvinhu nenzira yakatarwa kana mukurongeka kwakasiyana. Muchinyorwa chino, tichakurukura mienzaniso yakati wandei yematambudziko ane chekuita nemutemo wekuzadza nzvimbo, tichipa mhinduro dzakadzama dzechimwe nechimwe.
Pendauluan
Kuzadza nzvimbo inzira inoshandiswa zvakanyanya mu combinatorics, ndima yemasvomhu inodzidza kurongeka, kusanganiswa, uye kusarudzwa kwezvinhu. Imwe yenheyo huru dze combinatorics mutemo wekuwedzera, uyo unoti kana paine matanho akati wandei mukuita uye danho rega rega riine nhamba yakati yesarudzo, saka huwandu hwese hwekurongeka hunogona kuwanikwa nekuwedzera huwandu hwesarudzo mudanho rega rega.
Semuenzaniso, kana tine matanho maviri apo danho rekutanga rine sarudzo dze \(m\) uye danho rechipiri rine sarudzo dze \(n\), saka huwandu hwese hwezvirongwa zvinogoneka ndi \(m \times n\).
Ngatishandisei pfungwa iyi kugadzirisa mimwe mienzaniso yezvinetso.
Muenzaniso 1: Kuronga Mabhuku Pasherufu
Mubvunzo:
Kune mabhuku mashanu akasiyana uye sherufu yemabhuku ine nzvimbo shanu dzekuzadza. Mabhuku mashanu angarongwa sei pasherufu?
Kukurukurirana:
Muchiitiko ichi, tinofanira kuronga mabhuku mashanu munzvimbo shanu dzakasiyana. Iri idambudziko repermutation nekuti kurongeka kwakakosha. Tinogona kushandisa mutemo wekuzadza nzvimbo kana mutemo wekuwanza kuti tigadzirise dambudziko iri.
1. Mukamuri rekutanga, tine sarudzo shanu dzemabhuku.
2. Kana bhuku rimwe chete raiswa mukamuri rekutanga, tinosara nemabhuku mana ekusarudza ekamuri rechipiri.
3. Kune imba yechitatu, tine mabhuku matatu asara, nezvimwewo.
Equation yehuwandu hwese hwezvigadziro ndeiyi:
\[ 5 \kawa 4 \kawa 3 \kawa 2 \kawa 1 = 5! = 120 \]
Saka, kune nzira zana nemakumi maviri dzekuronga mabhuku mashanu.
Muenzaniso 2: Kugadzira Mashoko Kubva Mutsamba Dzakasiyana
Mubvunzo:
Mangani mazwi akasiyana anogona kuumbwa uchishandisa mavara ese ari mushoko rekuti "MATHEMATIKI", pasina kuadzokorora?
Kukurukurirana:
Tinofanira kutanga taona kuti mabhii mangani ari mushoko rekuti "MATHEMATIKI." Kune mabhii gumi nerimwe, mamwe acho anodzokororwa. Mabhii anodzokororwa ndeaya:
– M kusvika 2
- A kusvika ku3
– T kusvika 2
– Mamwe mavara (E, I, K) rimwe nerimwe rinoonekwa kamwe chete.
Tinoshandisa fomura ye permutation yezvinhu zvinodzokororwa, zvinoti:
\[ \frac{n!}{n_1! \times n_2! \times \ldots \times n_k!} \]
apo \( n \) iri nhamba yese yezvinhu (mavara) uye \( n_1, n_2, \ldots, n_k \) iri nhamba yekudzokororwa kwechinhu chimwe nechimwe chakasiyana.
Neshoko rekuti "MATAMBUDZIKO":
\[ n = 11, n_1 = 2 \text{ (M)}, n_2 = 3 \text{ (A)}, n_3 = 2 \text{ (T)}, n_4 = 1 \text{ (E)}, n_5 = 1 \text{ (I)}, n_6 = 1 \text{ (K)} \]
Saka huwandu hwemazwi anogona kuumbwa ndeinotevera:
\[ \frac{11!}{2! \nguva 3! \nguva 2! \nguva 1! \nguva 1! \nguva 1!} = \frac{39916800}{2 \nguva 6 \nguva 2 \nguva 1 \nguva 1 \nguva 1 \nguva 1} = \frac{39916800}{24} = 1663200 \]
Kune mazwi akasiyana-siyana anokwana 1,663,200 anogona kuumbwa.
Muenzaniso 3: Kuona Huwandu hweMisanganiswa muMartabak
Mubvunzo:
Mutengesi wemartabak anopa zvinhu zvishanu zvinozadzwa (chizi, chokoreti, nzungu, bhanana, nemazambiringa akaomeswa). Kana mutengi achida kusarudza zvitatu kubva pazvinhu zvishanu zvinozadzwa zvemartabak yake, angasarudza musanganiswa mingani yakasiyana?
Kukurukurirana:
Iri idambudziko remusanganiswa, kwete permutation, nekuti kurongeka kwacho hakuna kukosha. Isu tinoshandisa fomura yemusanganiswa:
\[ C(n, k) = \frac{n!}{k!(nk)!} \]
apo \( n \) iri nhamba yese yesarudzo, uye \( k \) iri nhamba yesarudzo dzakatorwa.
Panyaya iyi, \( n = 5 \) uye \( k = 3 \), saka:
\[ C(5, 3) = \frac{5!}{3!(5-3)!} = \frac{5!}{3! \times 2!} = \frac{120}{6 \times 2} = \frac{120}{12} = 10 \]
Kune misanganiswa gumi yakasiyana yekusarudza zviri mukati zvitatu kubva pane zvishanu.
Muenzaniso 4: Kurongeka kweVatori vechikamu mumutambo
Mubvunzo:
Kune vanhu vasere vanopinda mumakwikwi ekumhanya. Vatatu vepamusoro vanogona kuiswa munzira dzipi?
Kukurukurirana:
Iri idambudziko repermutation pasina kudzokorora nekuti chinzvimbo zvinoreva kuti kurongeka kwakakosha. Tinoshandisa fomura yepermutation:
\[ P(n, k) = \frac{n!}{(nk)!} \]
Panyaya iyi, \( n = 8 \) uye \( k = 3 \), zvino:
\[ P(8, 3) = \frac{8!}{(8-3)!} = \frac{8!}{5!} = \frac{40320}{120} = 336 \]
Saka, kune nzira mazana matatu nemakumi matatu nematanhatu dzekuisa nzvimbo nhatu dzepamusoro dzevatori vechikamu vasere.
Muchinyorwa chino, takurukura nezvematambudziko akati wandei nemhinduro dzawo tichishandisa mitemo yekuzadza nzvimbo mumamiriro akasiyana-siyana: kubva pakuronga mabhuku pasherufu kusvika pakuona anokunda mumakwikwi. Kunzwisisa izvi zvakakosha kuchakupa chivimbo chakawanda mukugadzirisa matambudziko akasiyana-siyana ekubatanidza uye mukana waungasangana nawo.
