Mafunso Okambirana a Combinatorics
Combinatorics ndi nthambi ya masamu yomwe imaphunzira kuwerengera, kukonza, ndi kapangidwe kake ka zinthu. Combinatorics ili ndi ntchito zofunika kwambiri m'magawo osiyanasiyana, kuphatikizapo sayansi ya makompyuta, ziwerengero, zamoyo, ndi zachuma. M'nkhaniyi, tikambirana zitsanzo zingapo ndi zokambirana zawo zokhudzana ndi combinatorics, zomwe zingapereke kumvetsetsa bwino mfundo zoyambira ndi momwe combinatorics imagwirira ntchito.
Funso 1: Kusasinthika
Funso:
Kodi mabuku 5 osiyanasiyana angakonzedwe m'njira zingati pa shelufu?
Kukambirana:
Kusinthasintha kwa zinthu ndi dongosolo la zinthu motsatira dongosolo. Ngati dongosolo ndi lofunika, timagwiritsa ntchito kusintha kwa zinthu. Pankhani ya vutoli, tili ndi mabuku asanu osiyanasiyana oti tikonze. Njira zingapo zokonzera mabuku asanu awa ndi izi:
\[ 5! = 5 \nthawi 4 \nthawi 3 \nthawi 2 \nthawi 1 = 120 \]
Kotero, pali njira 120 zokonzera mabuku 5 osiyanasiyana pa shelufu.
Funso 2: Kuphatikiza
Funso:
Kuchokera pa anthu 10, pali njira zingati zopangira gulu la anthu 4?
Kukambirana:
Kuphatikizana ndi kusankha zinthu zomwe dongosolo lake silili lofunika. Fomula yophatikizana ndi iyi:
\[ \binom{n}{k} = \frac{n!}{k!(nk)!} \]
Pankhani ya vutoli, \( n = 10 \) ndi \( k = 4 \). Chifukwa chake,
\[ \binom{10}{4} = \frac{10!}{4! \times (10-4)!} = \frac{10!}{4! \times 6!} \]
Tikudziwa kuti \( 10! = 10 \nthawi 9 \nthawi 8 \nthawi 7 \nthawi 6! \), ndiye
\[ \binom{10}{4} = \frac{10 \nthawi 9 \nthawi 8 \nthawi 7 \nthawi 6!}{4! \nthawi 6!} = \frac{10 \nthawi 9 \nthawi 8 \nthawi 7}{4 \nthawi 3 \nthawi 2 \nthawi 1} = 210 \]
Kotero, pali njira 210 zopangira gulu la anthu 4 mwa 10.
Funso 3: Zosintha ndi Kubwerezabwereza
Funso:
Kodi pali njira zingati zokonzera mawu oti “LEVEL”?
Kukambirana:
Mawu akuti “LEVEL” ali ndi zilembo 5, zina mwa izo zimabwerezedwa (L kawiri ndi E kawiri). Fomula yosinthira mawu ndi kubwerezabwereza ndi iyi:
\[ \frac{n!}{n_1! \times n_2! \times \ldots \times n_k!} \]
Pankhani ya vutoli, \( n = 5 \), \( n_1 = 2 \) ya chilembo L, ndi \( n_2 = 2 \) ya chilembo E. Chifukwa chake,
\[ \frac{5!}{2! \nthawi 2!} = \frac{5 \nthawi 4 \nthawi 3 \nthawi 2 \nthawi 1}{2 \nthawi 1 \nthawi 2 \nthawi 1} = \frac{120}{4} = 30 \]
Kotero, pali njira 30 zokonzera mawu oti “LEVEL”.
Funso 4: Kuphatikiza ndi Kubwerezabwereza
Funso:
Kodi pali njira zingati zosankhira maswiti atatu kuchokera ku mitundu 5 yosiyanasiyana ya maswiti omwe amaloledwa kubwerezabwereza?
Kukambirana:
Kuphatikiza ndi kubwerezabwereza pogwiritsa ntchito njira yotsatirayi:
\[ \binom{n+r-1}{r} \]
Pankhani ya vutoli, \( n = 5 \) (mitundu ya maswiti) ndi \( r = 3 \) (chiwerengero cha maswiti osankhidwa). Chifukwa chake,
\[ \binom{5+3-1}{3} = \binom{7}{3} = \frac{7!}{3! \times 4!} \]
Kudziwa \( 7! = 7 \nthawi 6 \nthawi 5 \nthawi 4! \), ndiye
\[ \binom{7}{3} = \frac{7 \nthawi 6 \nthawi 5 \nthawi 4!}{3! \nthawi 4!} = \frac{7 \nthawi 6 \nthawi 5}{3 \nthawi 2 \nthawi 1} = 35 \]
Kotero, pali njira 35 zosankhira maswiti atatu kuchokera ku mitundu 5 yosiyanasiyana ya maswiti omwe amaloledwa kubwerezabwereza.
Funso 5: Mfundo Yowonjezera
Funso:
Kodi pali njira zingati zosankhira chipatso chimodzi kuchokera m'basiketi yokhala ndi maapulo atatu, malalanje awiri, ndi nthochi zisanu?
Kukambirana:
Mfundo yowonjezerera imati ngati pali njira zingapo zochitira chinthu, ndiye kuti chiwerengero chonse cha njirazo ndi chiwerengero cha njira zonsezo. Potengera vutoli,
– Pali njira zitatu zosankhira apulo imodzi.
– Pali njira ziwiri zosankhira lalanje limodzi.
– Pali njira 5 zosankhira nthochi imodzi.
Njira zonse:
\[ 3 + 2 + 5 = 10 \]
Kotero, pali njira 10 zosankhira chipatso chimodzi kuchokera mumtanga.
Funso 6: Mfundo Yochulukitsa
Funso:
Kodi pali njira zingati zosankhira shati imodzi kuchokera ku mitundu inayi ndi thalauza limodzi kuchokera ku mitundu itatu?
Kukambirana:
Mfundo yochulukitsa imati ngati pali njira zingapo zochitira chinthu choyamba ndi njira zingapo zochitira chinthu chachiwiri, ndiye kuti njira zonse zochitira zinthu zonse ziwirizi ndi zotsatira za njira zochitira chinthu chilichonse.
Mu funso ili,
- Pali njira zinayi zosankhira shati imodzi.
- Pali njira zitatu zosankhira thalauza limodzi.
Njira zonse:
\[ 4 \nthawi 3 = 12 \]
Kotero, pali njira 12 zosankhira shati limodzi ndi thalauza limodzi.
Mapeto
Combinatorics, monga nthambi ya masamu, imapereka njira zambiri ndi malingaliro owerengera ndikukonza zinthu zosiyanasiyana. Kuyambira kusintha kwa zinthu mpaka mfundo zophatikiza ndi kuchulukitsa, malingaliro awa amagwiritsidwa ntchito nthawi zambiri m'njira zosiyanasiyana zothandiza. Pomvetsetsa zitsanzo ndi zokambirana zomwe zili pamwambapa, owerenga akuyembekezeka kugwiritsa ntchito malingaliro a combinatorics m'mikhalidwe yovuta kwambiri ndikukweza luso lawo lothetsa mavuto m'masamu ndi m'magawo ena.
