Zitsanzo za Mafunso Okhudza Malamulo Odzazira Malo
Lamulo lodzaza malo, kapena lamulo loyika, ndi lingaliro lofunikira mu masamu ndi kuthekera komwe ndi kothandiza kwambiri nthawi zambiri. Lamuloli nthawi zambiri limagwiritsidwa ntchito pokonza zinthu motsatira dongosolo linalake kapena m'makonzedwe osiyanasiyana. M'nkhaniyi, tikambirana zitsanzo zingapo za mavuto okhudzana ndi lamulo lodzaza malo, ndikupereka mayankho atsatanetsatane pa chilichonse.
Pendauluan
Kudzaza malo ndi njira yodziwika bwino yomwe imagwiritsidwa ntchito mu combinatorics, gawo la masamu lomwe limaphunzira za dongosolo, kuphatikiza, ndi kusankha zinthu. Chimodzi mwa mfundo zazikulu za combinatorics ndi lamulo lochulukitsa, lomwe limati ngati pali magawo angapo mu ndondomeko ndipo gawo lililonse lili ndi chiwerengero cha zisankho, ndiye kuti chiwerengero chonse cha makonzedwe omwe angatheke chingapezeke mwa kuchulukitsa chiwerengero cha zisankho mu gawo lililonse.
Mwachitsanzo, ngati tili ndi magawo awiri pomwe gawo loyamba lili ndi zosankha za \(m\) ndipo gawo lachiwiri lili ndi zosankha za \(n\), ndiye kuti chiwerengero chonse cha makonzedwe omwe angatheke ndi \(m \times n\).
Tiyeni tigwiritse ntchito lingaliro ili pothetsa mavuto ena.
Chitsanzo 1: Kukonza Mabuku Pa Shelufu
Funso:
Pali mabuku asanu osiyanasiyana ndi shelufu yokhala ndi malo asanu oti mudzaze. Kodi mabuku asanuwo angakonzedwe m'njira zingati pashelufu?
Kukambirana:
Pankhaniyi, tifunika kukonza mabuku asanu m'malo asanu osiyana. Ili ndi vuto la permutation chifukwa dongosolo ndilofunika kwambiri. Tingagwiritse ntchito lamulo lodzaza malo kapena lamulo lochulukitsa kuti tithetse vutoli.
1. Pa chipinda choyamba, tili ndi zosankha 5 za mabuku.
2. Buku limodzi litayikidwa m'chipinda choyamba, timatsala ndi mabuku anayi oti tisankhe m'chipinda chachiwiri.
3. Pa chipinda chachitatu, tili ndi zosankha zitatu zotsala za mabuku, ndi zina zotero.
Chiyerekezo cha chiwerengero chonse cha zoikamo ndi:
\[ 5 \nthawi 4 \nthawi 3 \nthawi 2 \nthawi 1 = 5! = 120 \]
Kotero, pali njira 120 zokonzera mabuku asanu.
Chitsanzo 2: Kupanga Mawu Kuchokera ku Zilembo Zosiyana
Funso:
Kodi ndi mawu angati osiyanasiyana omwe angapangidwe pogwiritsa ntchito zilembo zonse zomwe zili mu liwu lakuti "MASAMUKA", popanda kuwabwerezabwereza?
Kukambirana:
Choyamba tiyenera kuona kuti mawu oti "MASAMUKA" ali ndi zilembo zingati. Pali zilembo 11, zina mwa izo zimabwerezedwa. Zilembo zobwerezedwa ndi izi:
– M mpaka 2
- A mpaka 3
– T mpaka 2
– Zilembo zina (E, I, K) chilichonse chimawonekera kamodzi.
Timagwiritsa ntchito njira yosinthira zinthu pa zinthu zobwerezabwereza, zomwe ndi:
\[ \frac{n!}{n_1! \times n_2! \times \ldots \times n_k!} \]
kumene \( n \) ndi chiwerengero chonse cha zinthu (zilembo) ndipo \( n_1, n_2, \ldots, n_k \) ndi chiwerengero cha kubwerezabwereza kwa chinthu chilichonse chosiyana.
Ndi mawu oti "MASAMUKA":
\[ n = 11, n_1 = 2 \malemba{ (M)}, n_2 = 3 \malemba{ (A)}, n_3 = 2 \malemba{ (T)}, n_4 = 1 \malemba{ (E)}, n_5 = 1 \malemba{ (I)}, n_6 = 1 \malemba{ (K)} \]
Kotero chiwerengero cha mawu omwe angapangidwe ndi awa:
\[ \frac{11!}{2! \nthawi 3! \nthawi 2! \nthawi 1! \nthawi 1! \nthawi 1!} = \frac{39916800}{2 \nthawi 6 \nthawi 2 \nthawi 1 \nthawi 1 \nthawi 1 \nthawi 1} = \frac{39916800}{24} = 1663200 \]
Pali mawu osiyanasiyana okwana 1,663,200 omwe angapangidwe.
Chitsanzo 3: Kudziwa Chiwerengero cha Zosakaniza mu Martabak
Funso:
Wogulitsa martabak amapereka njira zisanu zodzazira (tchizi, chokoleti, mtedza, nthochi, ndi mphesa zouma). Ngati kasitomala akufuna kusankha zitatu mwa zisanu zodzazira martabak yake, ndi mitundu ingati yosiyanasiyana yomwe angasankhe?
Kukambirana:
Ili ndi vuto lophatikizana, osati permutation, chifukwa dongosolo lake silofunika kwenikweni. Timagwiritsa ntchito njira yophatikizana:
\[ C(n, k) = \frac{n!}{k!(nk)!} \]
kumene \( n \) ndi chiwerengero chonse cha zisankho, ndipo \( k \) ndi chiwerengero cha zisankho zomwe zatengedwa.
Pachifukwa ichi, \( n = 5 \) ndi \( k = 3 \), kotero:
\[ C(5, 3) = \frac{5!}{3!(5-3)!} = \frac{5!}{3! \times 2!} = \frac{120}{6 \times 2} = \frac{120}{12} = 10 \]
Pali mitundu 10 yosiyanasiyana yosankha zinthu zitatu kuchokera pa mitundu 5.
Chitsanzo 4: Kukonzekera kwa Otenga nawo mbali pa mpikisano
Funso:
Pali anthu 8 omwe akutenga nawo mbali mu mpikisano wothamanga. Kodi anthu atatu apamwamba angaikidwe m'njira zingati?
Kukambirana:
Ili ndi vuto la permutation popanda kubwerezabwereza chifukwa malo amatanthauza kuti dongosolo ndilofunika. Timagwiritsa ntchito njira ya permutation:
\[ P(n, k) = \frac{n!}{(nk)!} \]
Pachifukwa ichi, \( n = 8 \) ndi \( k = 3 \), ndiye:
\[ P(8, 3) = \frac{8!}{(8-3)!} = \frac{8!}{5!} = \frac{40320}{120} = 336 \]
Kotero, pali njira 336 zoyikira malo atatu apamwamba a ophunzira 8.
Munkhaniyi, takambirana zitsanzo zingapo za mavuto ndi mayankho awo pogwiritsa ntchito malamulo odzaza malo m'mikhalidwe yosiyanasiyana: kuyambira kukonza mabuku pashelefu mpaka kudziwa wopambana mpikisano. Kumvetsetsa mfundo izi kukupatsani chidaliro chowonjezereka pakuthana ndi mavuto osiyanasiyana okhudzana ndi kuphatikizana ndi kuthekera komwe mungakumane nako.
