Mibvunzo yekukurukurirana yeCombinatorics
Combinatorics ibazi remasvomhu rinoongorora kuverenga, kurongeka, uye maumbirwo angangodaro ezvikamu zvezvinhu. Combinatorics ine mashandisirwo akakosha muzvikamu zvakasiyana-siyana, zvinosanganisira sainzi yemakombiyuta, nhamba, biology, uye economics. Muchinyorwa chino, tichakurukura mienzaniso yakati wandei nehurukuro dzadzo dzine chekuita necombinatorics, izvo zvinotarisirwa kuti zvichapa kunzwisisa kuri nani kwepfungwa huru uye mashandisirwo ecombinatorics.
Mubvunzo 1: Kuchinja-chinja
Mubvunzo:
Mabhuku mashanu akasiyana angarongwa sei pasherufu?
Kukurukurirana:
Kurongeka kwezvinhu nenzira yakarongeka. Kana kurongeka kwakakosha, tinoshandisa kurongeka. Munyaya yedambudziko iri, tine mabhuku mashanu akasiyana ekuronga. Nzira dzekuronga mabhuku mashanu aya ndeidzi:
\[ 5! = 5 \kawa 4 \kawa 3 \kawa 2 \kawa 1 = 120 \]
Saka, kune nzira zana nemakumi maviri dzekuronga mabhuku mashanu akasiyana pasherufu.
Mubvunzo 2: Musanganiswa
Mubvunzo:
Kubva pavanhu gumi, nzira ngani dziripo dzekuumba timu yevanhu vana?
Kukurukurirana:
Kusanganiswa kusarudzwa kwezvinhu uko kurongeka kusina kukosha. Fomura yekubatanidza ndeiyi:
\[ \bhinom{n}{k} = \frac{n!}{k!(nk)!} \]
Munyaya yedambudziko iri, \( n = 10 \) uye \( k = 4 \). Saka,
\[ \binom{10}{4} = \frac{10!}{4! \times (10-4)!} = \frac{10!}{4! \times 6!} \]
Tinoziva kuti \( 10! = 10 \kawa 9 \kawa 8 \kawa 7 \kawa 6! \), zvino
\[ \binom{10}{4} = \frac{10 \nguva 9 \nguva 8 \nguva 7 \nguva 6!}{4! \nguva 6!} = \frac{10 \nguva 9 \nguva 8 \nguva 7}{4 \nguva 3 \nguva 2 \nguva 1} = 210 \]
Saka, kune nzira mazana maviri negumi dzekuumba timu ine vanhu vana pagumi.
Mubvunzo 3: Kuchinja-chinja nekudzokorora
Mubvunzo:
Pane nzira ngani dzekuronga izwi rekuti “LEVEL”?
Kukurukurirana:
Izwi rekuti “LEVEL” rine mabhii mashanu, mamwe acho anodzokororwa (L kaviri uye E kaviri). Fomura yekuchinja ine kudzokorora ndeiyi:
\[ \frac{n!}{n_1! \times n_2! \times \ldots \times n_k!} \]
Munyaya yedambudziko iri, \( n = 5 \), \( n_1 = 2 \) yebhii L, uye \( n_2 = 2 \) yebhii E. Saka,
\[ \frac{5!}{2! \times 2!} = \frac{5 \times 4 \times 3 \times 2 \times 1}{2 \times 1 \times 2 \times 1} = \frac{120}{4} = 30 \]
Saka, kune nzira makumi matatu dzekuronga izwi rekuti "LEVEL".
Mubvunzo 4: Kusanganiswa neKudzokorora
Mubvunzo:
Pane nzira ngani dzekusarudza masiwiti matatu kubva mumhando shanu dzakasiyana dzemasiwiti uye kudzokorora kunobvumidzwa?
Kukurukurirana:
Kusanganiswa nekudzokorora uchishandisa fomura inotevera:
\[ \binom{n+r-1}{r} \]
Munyaya yedambudziko iri, \( n = 5 \) (mhando dzemasiwiti) uye \( r = 3 \) (nhamba yemasiwiti akasarudzwa). Saka,
\[ \binom{5+3-1}{3} = \binom{7}{3} = \frac{7!}{3! \kawa 4!} \]
Kuziva \( 7! = 7 \kawa 6 \kawa 5 \kawa 4! \), wobva
\[ \binom{7}{3} = \frac{7 \kawa 6 \kawa 5 \kawa 4!}{3! \kawa 4!} = \frac{7 \kawa 6 \kawa 5}{3 \kawa 2 \kawa 1} = 35 \]
Saka, kune nzira makumi matatu neshanu dzekusarudza masiwiti matatu kubva mumhando shanu dzakasiyana dzemasiwiti ane kudzokorora kunobvumirwa.
Mubvunzo 5: Nheyo yekuwedzera
Mubvunzo:
Pane nzira ngani dzekusarudza muchero mumwe chete kubva mubhasikiti rine maapuro matatu, maranjisi maviri, uye mabhanana mashanu?
Kukurukurirana:
Nheyo yekuwedzera inoti kana paine nzira dzakawanda dzekuita chiito, saka huwandu hwenzira dzose ihwohwo huwandu hwenzira idzodzo dzose. Munyaya yedambudziko iri,
- Kune nzira nhatu dzekusarudza apuro rimwe chete.
- Kune nzira mbiri dzekusarudza orenji rimwe.
- Kune nzira shanu dzekusarudza bhanana rimwe chete.
Nzira dzese:
\[ 3 + 2 + 5 = 10 \]
Saka, kune nzira gumi dzekusarudza muchero mumwe chete kubva mudengu.
Mubvunzo 6: Nheyo yeKuwanza
Mubvunzo:
Pane nzira dzingani dzekusarudza hembe imwe chete kubva muzvisarudzo zvina uye bhurugwa rimwe kubva muzvisarudzo zvitatu?
Kukurukurirana:
Nheyo yekuwanda inoti kana paine nzira dzakawanda dzekuita chiito chekutanga uye nzira dzakawanda dzekuita chiito chechipiri, saka huwandu hwese hwenzira dzese dzekuita zviito zvese izvi chibereko chenzira dzekuita chiito chimwe nechimwe.
Mumubvunzo uyu,
- Kune nzira ina dzekusarudza hembe imwe chete.
- Kune nzira nhatu dzekusarudza bhurugwa rimwe chete.
Nzira dzese:
\[ 4 \kawa 3 = 12 \]
Saka, kune nzira gumi nembiri dzekusarudza shati rimwe chete nebhurugwa rimwe chete.
Mhedziso
Combinatorics, sebazi remasvomhu, inopa nzira dzakasiyana-siyana dzekuverenga nekuronga zvinhu zvakasiyana-siyana. Kubva pakuchinjika uye kusanganiswa kusvika pamisimboti yekuwedzera nekuwanda, pfungwa idzi dzinowanzo shandiswa mumhando dzakasiyana dzemashandisirwo. Nekunzwisisa mienzaniso nehurukuro dziri pamusoro, zvinotarisirwa kuti vaverengi vachakwanisa kushandisa pfungwa dzecombinatorics mumamiriro ezvinhu akaomarara uye kuvandudza hunyanzvi hwavo hwekugadzirisa matambudziko mumasvomhu nedzimwe nzvimbo.
