Muenzaniso weMibvunzo yeKukurukurirana Yakasanganiswa
Kusanganiswa ipfungwa inokosha mudzidziso yekufungidzira uye nhamba, zvichitibvumira kuverenga huwandu hwenzira dzekusarudza zvinhu kubva muchikamu chimwe chete pasina kutarisa kurongeka. Mukubatanidzwa, kurongeka kwekusarudza kunoramba kusina kuchinja, kusiyana nekubvumidzwa, uko kurongeka kwekusarudza kwakakosha. Kusanganiswa kunoshandiswa muminda yakasiyana-siyana, kubva kusainzi yekombuta nebiology kusvika kuhupfumi nemasvomhu. Chinyorwa chino chichatsanangura pfungwa huru yekubatanidza uye kupa mienzaniso yakati wandei yezvinetso nehurukuro kuti zvikubatsire kunzwisisa.
Pfungwa Yekutanga Yekusanganiswa
Tisati taenderera mberi nemibvunzo yemuenzaniso, ngatitangei nekunzwisisa kwekutanga kwemisanganiswa.
Kusanganiswa kwezvinhu \(n\) zvakatorwa \(r\) panguva imwe chete kunoratidzwa nechinyorwa \(\binom{n}{r}\) kana \(\mathbf{C}(n, r)\). Fomura yekuverenga musanganiswa ndeiyi:
\[ \bhinom{n}{r} = \frac{n!}{r!(nr)!} \]
– \( n \) ndiyo nhamba yezvinhu zviripo.
– \( r \) inhamba yezvinhu zvinofanira kusarudzwa.
– ! (factorial) ibasa remasvomhu rinowanza nhamba nenhamba dzose dzakanaka dziri pasi payo kusvika pa1.
Mibvunzo yemuenzaniso nekukurukurirana
Muenzaniso Mubvunzo 1: Kusarudza Chikwata Kubva muBoka reVadzidzi
Mubvunzo:
Mukirasi imwe chete mune vadzidzi gumi. Vadzidzi vana kubva mukirasi imwe chete vangasarudzwa sei kuti vatore chikamu mumakwikwi?
Kukurukurirana:
Kushandisa fomura yekubatanidza:
\[ \binom{10}{4} = \frac{10!}{4! \cdot (10-4)!} = \frac{10!}{4! \cdot 6!} \]
Kuverenga factorial:
– \( 10! = 10 \kawa 9 \kawa 8 \kawa 7 \kawa 6! \)
– \( 4! = 4 \kawa 3 \kawa 2 \kawa 1 = 24 \)
– \( 6! \) yatove pamusoro, saka inogona kudzima 6! mu \(10!\).
Saka:
\[ \binom{10}{4} = \frac{10 \nguva 9 \nguva 8 \nguva 7}{4 \nguva 3 \nguva 2 \nguva 1} = \frac{5040}{24} = 210 \]
Saka, kune nzira mazana maviri negumi dzekusarudza vadzidzi vana kubva kuvadzidzi gumi.
Muenzaniso Mubvunzo 2: Kusanganiswa Kunoshandisa Makadhi
Mubvunzo:
Kubva padheki rine makadhi makumi mashanu nemaviri, nzira ngani dziripo dzekusarudza ruoko rwepoker rwemakadhi mashanu?
Kukurukurirana:
Kushandisa fomura yekubatanidza:
\[ \binom{52}{5} = \frac{52!}{5! \cdot (52-5)!} = \frac{52!}{5! \cdot 47!} \]
Kuverenga factorial:
– \( 52! = 52 \kawa 51 \kawa 50 \kawa 49 \kawa 48 \kawa 47! \)
– \( 5! = 5 \kawa 4 \kawa 3 \kawa 2 \kawa 1 = 120 \)
– \( 47! \) yatove pamusoro, saka inogona kudzima 47! mu \(52!\).
Saka:
\[ \binom{52}{5} = \frac{52 \nguva 51 \nguva 50 \nguva 49 \nguva 48}{5 \nguva 4 \nguva 3 \nguva 2 \nguva 1} = \frac{311875200}{120} = 2598960 \]
Saka, kune nzira dzinosvika 2,598,960 dzekusarudza makadhi mashanu kubva pamakadhi makumi mashanu nemaviri.
Muenzaniso Mubvunzo 3: Kuverenga Misanganiswa Yekusarudza Chikwata Chemitambo
Mubvunzo:
Chikwata chebasketball chine vatambi gumi nevashanu. Kana mudzidzisi achida kusarudza vatambi vashanu sevatambi vekutanga, izvi zvingaitwa nenzira dzipi?
Kukurukurirana:
Kushandisa fomura yekubatanidza:
\[ \binom{15}{5} = \frac{15!}{5! \cdot (15-5)!} = \frac{15!}{5! \cdot 10!} \]
Kuverenga factorial:
– \( 15! = 15 \kawa 14 \kawa 13 \kawa 12 \kawa 11 \kawa 10! \)
– \( 5! = 5 \kawa 4 \kawa 3 \kawa 2 \kawa 1 = 120 \)
– \( 10! \) yatove pamusoro, saka inogona kudzima 10! mu \(15!\).
Saka:
\[ \binom{15}{5} = \frac{15 \nguva 14 \nguva 13 \nguva 12 \nguva 11}{5 \nguva 4 \nguva 3 \nguva 2 \nguva 1} = \frac{360360}{120} = 3003 \]
Saka, kune nzira dzinosvika zviuru zvitatu nenhatu dzekusarudza vatambi vashanu kubva pavatambi gumi nevashanu.
Muenzaniso Mubvunzo 4: Musanganiswa muBiology
Mubvunzo:
Mubindu, mune marudzi masere akasiyana emaruva. Marudzi matatu emaruva angasarudzwa sei kuti pave nemaruva akarongwa?
Kukurukurirana:
Kushandisa fomura yekubatanidza:
\[ \binom{8}{3} = \frac{8!}{3! \cdot (8-3)!} = \frac{8!}{3! \cdot 5!} \]
Kuverenga factorial:
– \( 8! = 8 \kawa 7 \kawa 6 \kawa 5! \)
– \( 3! = 3 \kawa 2 \kawa 1 = 6 \)
– \( 5! \) yatove pamusoro, saka inogona kudzima 5! mu \(8!\).
Saka:
\[ \binom{8}{3} = \frac{8 \times 7 \times 6}{3 \times 2 \times 1} = \frac{336}{6} = 56 \]
Saka, kune nzira makumi mashanu nematanhatu dzekusarudza mhando nhatu dzemaruva kubva mumhando sere dzemaruva.
Muenzaniso Mubvunzo 5: Kusanganiswa mukuvaka chikwata
Mubvunzo:
Une mabhuku gumi nemaviri akasiyana pasherufu yemabhuku. Ungasarudza sei mabhuku mashanu kubva pamabhuku gumi nemaviri nenzira ngani?
Kukurukurirana:
Kushandisa fomura yekubatanidza:
\[ \binom{12}{5} = \frac{12!}{5! \cdot (12-5)!} = \frac{12!}{5! \cdot 7!} \]
Kuverenga factorial:
– \( 12! = 12 \kawa 11 \kawa 10 \kawa 9 \kawa 8 \kawa 7! \)
– \( 5! = 5 \kawa 4 \kawa 3 \kawa 2 \kawa 1 = 120 \)
– \( 7! \) yatove pamusoro, saka inogona kudzima 7! mu \(12!\).
Saka:
\[ \binom{12}{5} = \frac{12 \nguva 11 \nguva 10 \nguva 9 \nguva 8}{5 \nguva 4 \nguva 3 \nguva 2 \nguva 1} = \frac{95040}{120} = 792 \]
Saka, kune nzira dzinosvika mazana manomwe nemakumi mapfumbamwe nembiri dzekusarudza mabhuku mashanu kubva mumabhuku gumi nemaviri.
Mhedziso
Kusanganiswa inzira inobatsira zvikuru muminda yakasiyana-siyana, kunyanya kana tichida kuziva huwandu hwenzira dzekusarudza zvinhu tisingatarisi kurongeka. Tichishandisa fomura yekutanga yekubatanidza \(\binom{n}{r}\), tinogona kuzviverenga nekukurumidza uye zviri nyore, chero bedzi tichinzwisisa mashandiro anoita factorials. Kuburikidza nemienzaniso iri pamusoro, tinotarisira kuva nekunzwisisa kwakakwana uye kwakajeka kwekubatanidza. Ramba uchidzidzira nematambudziko akasiyana-siyana kuti uwedzere kusimbisa kunzwisisa kwako.
