Mitemo yePermutation uye yeMusanganiswa
Mumasvomhu, kunyanya munyaya dzezvingangoitika uye nhamba, tinowanzo sangana nemubvunzo wekuti "nzira ngani" chiitiko chinogona kuitika. Semuenzaniso: pane nzira ngani dzekugara dzevanhu dzakatarwa? Nhengo dzechikwata dzingasarudzwa sei kubva muboka revadzidzi? Mibvunzo yakaita seiyi inopindurwa uchishandisa mitemo yepermutation ne combination, pfungwa mbiri huru mumitemo yekuverenga. Kunyange zvazvo dzose dzichibata ne "nhamba yenzira," musiyano wakakosha uri pakuti kurongeka kunofungwa here.
1. Pfungwa huru yeMitemo yeKunyora
Tisati tanyatsoongorora ma permutations uye musanganiswa, pane pfungwa huru yekunzwisisa: kuverenga inzira yekuverenga huwandu hwemhedzisiro inogona kuitika kubva pane imwe mamiriro ezvinhu. Kuverenga kunogona kuitwa nemaoko kune zviitiko zvidiki, asi kune zviitiko zvikuru, tinoda fomura inoshanda.
Nheyo mbiri huru mukuverengwa kwenhamba ndeidzi:
1. Mutemo weChigadzirwa
Kana maitiro aine matanho akati wandei, uye danho rega rega riine sarudzo dzakasiyana-siyana, saka huwandu hwenzira dzese hunobva muhuwandu hwesarudzo padanho rega rega.
2. Mutemo weSum
Kana sarudzo ikagona kuitwa nenzira dzakasiyana-siyana (dzisingapindirane), saka huwandu hwenzira idzodzo ndihwo huwandu hwenzira idzodzo.
Kuchinja-chinja uye kusanganiswa kwezvishandiso zvepfungwa iyi, kunyanya kana zvinhu zvatanga kurongwa kana kusarudzwa.
2. Kutenderwa: Kurongeka Nekuteerera Kurongeka
Permutation inzira yekuronga kana kusarudza zvinhu pazvinonyanya kukosha pakurongeka. Izvi zvinoreva kuti kurongeka kweAB kwakasiyana nekweBA.
a. Kuchinja kwezvinhu zvakasiyana-siyana (zvese zvakarongwa)
Kana paine zvinhu zvakasiyana-siyana zvinofanira kurongwa zvese munhevedzano, nhamba yekurongwa ndeiyi:
\[
n! = n \nguva (n-1) \nguva (n-2) \nguva \madotsi \nguva 2 \nguva 1
\]
Chiratidzo chekuti “!” chinonzi factorial.
Muenzaniso:
Kune mabhuku mana akasiyana. Mabhuku aya anogona kurongwa sei pasherufu?
\[
4! = 4 \kawa 3 \kawa 2 \kawa 1 = 24
\]
Saka kune zvirongwa makumi maviri nemana.
b. Kuchinja kwechikamu: kusarudza r kubva kuna n (kurongeka kunotariswa)
Kana tikasarudza zvinhu r kuti tironge kubva pazvinhu zvakasiyana-siyana kubva ku n (kwete zvese), saka fomura ye permutation ndeiyi:
\[
P(n,r) = \frac{n!}{(nr)!}
\]
Muenzaniso:
Pavadzidzi vatanhatu, vadzidzi vatatu vachasarudzwa kuti vave sachigaro, mutevedzeri wesachigaro, uye munyori. Izvi zvingaitwa munzira dzipi?
Sezvo sachigaro-mutevedzeri wemunyori vari zvinzvimbo zvakasiyana, kurongeka kwacho kwakakosha.
\[
P(6,3) = \frac{6!}{(6-3)!}=\frac{6!}{3!} = 6 \kawa 5 \kawa 4 = 120
\]
Kune nzira dzinosvika zana nemakumi maviri.
c. Kuchinja-chinja kwechinhu chimwe chete (kudzokorora/kufanana)
Dzimwe nguva pane zvinhu zvisingasiyane zvachose. Semuenzaniso, mushoko rekuti "USIKU" pane maM maviri nemaA maviri (kana kuti "USIKU": pane maM maviri, pane maA maviri? Kutaura zvazviri, "USIKU" = USIKU: M=2, A=2, L=1). Huwandu hwemarongerwo akasiyana hunoverengerwa ne:
\[
\frac{n!}{n_1! \, n_2! \, \dots}
\]
apo \(n\) iri nhamba yose yezvinhu, uye \(n_1, n_2\) iri nhamba yezvinhu zvakafanana.
Muenzaniso:
Marongerwo akasiyana-siyana emabhii ari mu "NIGHT" angangani?
Nhamba yemabhii \(n=5\), M ane 2, A ane 2, L ane 1.
\[
\frac{5!}{2!\,2!} = \frac{120}{4} = 30
\]
Saka kune marongero makumi matatu akasiyana.
3. Musanganiswa: Sarudzo Pasina Kutarisa Kurongeka
Kusanganisa inzira yekusarudza zvinhu pasina kurongeka kwazvinoita. Kusarudza A naB kwakafanana nekusarudza B naA.
Fomura yekubatanidza inosarudza r kubva kuzvinhu n:
\[
C(n,r) = \binom{n}{r}=\frac{n!}{r!(nr)!}
\]
a. Muenzaniso wemusanganiswa uri nyore
Muenzaniso:
Pavadzidzi gumi, vadzidzi vatatu vachasarudzwa kuti vave nhengo dzechikwata chemakwikwi (vasina mabasa chaiwo). Munzira dzipi?
Sezvo pasina ma ranks, kurongeka kwacho hakuna kukosha.
\[
C(10,3)=\frac{10!}{3!\,7!}=\frac{10 \kawa 9 \kawa 8}{3 \kawa 2 \kawa 1}=120
\]
Kune nzira dzinosvika zana nemakumi maviri.
b. Hukama huripo pakati pema permutations ne ma combinations
Cherechedza kuti ma permutations nema combinations zvine hukama. Kuti tisarudze vanhu r uye tivaronge, tinogona:
– tanga wasarudza vanhu vari mu "r": \(C(n,r)\)
- ronga r munhu iyeye: \(r!\)
Kuti:
\[
P(n,r) = C(n,r)\nguva r!
\]
Izvi zvinoratidza kuti permutation "yakakura" nekuti inosiyanisa kurongeka.
4. Maitiro Ekuziva: Shandisa Permutation kana Musanganiswa?
Kuti ugadzirise dambudziko, danho rakakosha nderekuziva kana kurongeka kwacho kuchifungwa nezvako.
Shandisa ma permutations kana:
– pane chinzvimbo kana chinzvimbo (sachigaro, mutevedzeri, nzvimbo 1-2-3),
- pane nzvimbo yekugara,
- pane kodhi kana kurongeka kwehurongwa.
Shandisa musanganiswa kana:
- sarudza nhengo dzeboka chete,
- kurongeka hakusiyanisi mhedzisiro,
- chinonyanya kukosha ndechekuti ndiani anosarudzwa, kwete chinzvimbo chavo.
Muenzaniso unokurumidza:
– Sarudza vanhu vashanu kubva pagumi nevaviri kuti vave mukomiti: musanganiswa
- Kusarudza vanokunda vekutanga, vechipiri, uye vechitatu kubva muvatori vechikamu gumi nevaviri: permutation
5. Mienzaniso yeMashandisirwo Muhupenyu Hwezuva Nezuva
Kuchinja-chinja uye kusanganiswa kwezvinhu hazvingoonekwi mumabhuku emasvomhu chete, asiwo mumamiriro ezvinhu chaiwo:
1. Kuchengetedzwa kwepassword (password/PIN)
Nhamba yemaPIN ane manhamba mana (0–9) anogona kunge aine kudzokorora kunobvumirwa ndeye \(10^4\). Izvi zvine chekuita nemutemo wekuwedzera uye pfungwa yekuchinja-chinja nekudzokorora.
2. Kuronga purogiramu kana zvigaro
Kusarudza nzvimbo dzekugara muzviitiko zvepamutemo uchishandisa ma permutations nekuda kwenzvimbo dzakasiyana.
3. Kusarudzwa kwechikwata kana komiti
Kusarudza vanhu vakati wandei kubva muboka rimwe chete kunosanganisira, nekuti kurongeka kwacho hakuna kukosha.
4. Mitambo yemakadhi
Misanganiswa inowanzo shandiswa kuverenga mukana wekuti ruoko rwakati rupiwe mupoker kana mimwe mitambo.
6. Zvikanganiso Zvakajairika Zvekudzivirira
Zvimwe zvikanganiso zvinowanzoitika pakushanda pamatambudziko ekugadzirisa uye ekubatanidza:
– Kufunga kuti kurongeka uku hakuna kukosha kunyangwe kuchikosha, semuenzaniso kusarudza sachigaro nemutevedzeri wake (kunofanira kunge kuri kutenderwa).
– Kukanganwa kupatsanura zvinhu zvakafanana, zvakaita sekunyora mazwi ane mavara akadzokororwa.
– Kuverenga zvisirizvo ma factorials, kunyanya pakurerutsa chimiro \(\frac{n!}{(nr)!}\).
Imwe nzira yekudzivirira izvi ndeyekunyora dudziro yemubvunzo mumitsara iri nyore: “Ndinosarudza here kana kuti ndinoronga here?” uye “Chinzvimbo chinoita mutsauko here mumhedzisiro?”
Penutup
Mitemo yekubvumidza uye kusanganisa zvinhu zvakakosha pakuverenga huwandu hwezvinogona kuitika mumamiriro akasiyana-siyana. Kubvumidza kunoshandiswa kana kurongeka kana chinzvimbo kuchikosha, nepo kusanganiswa kuchishandiswa kana kurongeka kusina kukosha. Nekunzwisisa musiyano uyu, kuziva factorials, uye kushandisa mafomula akakodzera, tinogona kugadzirisa matambudziko mazhinji ekuverenga uye mukana nekukurumidza uye nemazvo. Mukuita, kugona kusarudza nzira chaiyo—kubvumidza kana kusanganisa—kazhinji kwakakosha pane kungoyeuka mafomula.
