Motsoako
Pendahuluan
Lipalong tsa lipalo, motsoako ke mohopolo oa motheo o sebelisoang ho fumana palo ea litsela tsa ho khetha sete ea lintho ntle le ho nahana ka tatellano ea tsona. Metsoako e sebelisoa khafetsa mafapheng a fapaneng a mahlale, ho kenyeletsoa lipalo-palo, monyetla le k'homphieutha, hammoho le bophelong ba letsatsi le letsatsi, joalo ka kemiso, lipapali tsa likarete le tse ling. Sengoloa sena se tla hlahloba mohopolo oa metsoako ka botebo, mekhoa e sebelisitsoeng, le tse ling tsa lits'ebetso tsa tsona tse sebetsang.
Tlhaloso ea Motsoako
Motswako ke mokhoa oa ho khetha lintho tse 'maloa ho tsoa sete ntle le ho nahana ka tatellano eo li khethiloeng ka eona. Haeba ho na le lintho tse fapaneng tse \( n \) 'me re batla ho khetha lintho tse \( r \) ho tsoa sete, joale motsoako o ka hlalosoa ka mongolo o \( C(n, r) \) kapa \( \binom{n}{r} \).
Foromo ea Motsoako
Foromo ea ho bala palo ea metsoako \( \binom{n}{r} \) ke:
\[ \binom{n}{r} = \frac{n!}{r! (nr)!} \]
Moo \( n! \) (bala “n factorial”) e leng sehlahisoa sa linomoro tsohle tse ntle ho fihlela ho \( n \). Mohlala, \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \).
Mohlala oa Palo ea Motsoako
A re re re na le lintho tse 5 tse fapaneng, 'me re batla ho khetha lintho tse 3 ho tsoa sete. Re ka etsa see ka litsela tse kae?
Ho sebelisa foromo ea motsoako:
\[ \binom{5}{3} = \frac{5!}{3! (5-3)!} = \frac{5!}{3! \cdot 2!} = \frac{120}{6 \cdot 2} = 10 \]
Kahoo, ho na le litsela tse 10 tsa ho khetha lintho tse 3 ho tse 5 ntle le ho ela hloko tatellano.
Metswako ea Monyetla
Hangata metsoako e sebediswa ho fumana kgonahalo ya ketsahalo. Mohlala, dipapading tsa dikarete, re ka sebedisa metsoako ho bala kgonahalo ya ho fumana letsoho le itseng.
Mohlala oa 1: Papali ea Likarete
A re re re batla ho bala monyetla oa ho fumana para ea li-Ace ka likarete tse hlano tse sebelisoang ho tsoa mokatong o tloaelehileng oa likarete tse 52. Re ka sebelisa metsoako ho bala palo eohle ea litsela tsa ho khetha likarete tse hlano ho tsoa likarete tse 52 le palo ea litsela tsa ho khetha li-Ace tse peli ho tsoa ho li-Ace tse 'ne.
Litsela tsohle tsa ho khetha likarete tse 5 ho tse 52:
\[ \binom{52}{5} = \frac{52!}{5! \cdot 47!} \]
Mokhoa oa ho khetha li-Ace tse 2 ho li-Ace tse 4:
\[ \binom{4}{2} = \frac{4!}{2! \cdot 2!} = 6 \]
Mokhoa oa ho khetha likarete tse 3 ho tsoa likareteng tse 48 tse setseng (kaha likarete tse 2 li se li khethiloe ho tsoa ho li-Ace tse 4):
\[ \binom{48}{3} = \frac{48!}{3! \cdot 45!} \]
Kahoo, monyetla oa ho fumana li-Ace tse peli ka letsoho la likarete tse hlano ke:
\[ P = \frac{\binom{4}{2} \cdot \binom{48}{3}}{\binom{52}{5}} \]
Metswako ho Lipalopalo
Lipalopalong, metsoako e sebelisoa litlhahlobong tse fapaneng, ho kenyeletsoa le ho etsa liteko tsa ho nka lisampole le tsa khopolo-taba. Metswako e thusa ho fumana palo ea litsela tsa ho khetha sampole e itseng ho tsoa ho baahi.
Mohlala oa 2: Ho etsa sampole
A re re re na le baahi ba 20 mme re batla ho kgetha sampole ya batho ba 4. Re ka sebedisa metswako ho fumana palo ya ditsela tsa ho kgetha sampole eo.
\[ \binom{20}{4} = \frac{20!}{4! \cdot 16!} \]
Palo ena e fana ka tlhahisoleseding ea bohlokoa litlhahlobong tse fapaneng tsa lipalo-palo, joalo ka lipalo-palo tsa menyetla kapa khakanyo ea liparamente.
Khopolo-taba ea Motsoako le Palo
Metswako e boetse e bapala karolo ea bohlokoa khopolo-taba ea linomoro, haholo-holo moelelong oa li-coefficient tsa binomial le boitsebiso bo bong ba lipalo. Mohlala, Binomial Theorem e sebelisa metswako ho nts'etsapele lipolelo tsa algebra.
Theorem ea Binomial
Theorem ea Binomial e re:
\[ (x + y)^n = \ kakaretso_{k=0}^{n} \binom{n}{k} x^{nk} y^k \]
Mona, coefficient ea binomial \( \binom{n}{k} \) e bontša palo ea litsela tsa ho khetha lintho \( k \) ho tsoa linthong \( n \).
Litšebeliso tsa Motsoako Bophelong ba Letsatsi le Letsatsi
Khopolo ea ho kopanya ha e felle feela lipalo tsa khopolo-taba, empa hape e na le lits'ebetso tse ngata tse sebetsang bophelong ba letsatsi le letsatsi.
Mohlala oa 3: Litlhophiso tsa Kemiso
A re re re na le mesebetsi e 5 eo re lokelang ho e phetha ka beke, 'me re batla ho khetha e 3 eo re tla e sebetsa ka Mantaha. Re ka sebelisa metsoako ho fumana hore na re ka khetha mesebetsi e 3 ho tsoa ho e 5 eo ka palo.
\[ \binom{5}{3} = 10 \]
Kahoo, ho na le litsela tse 10 tsa ho hlophisa kemiso.
Mohlala oa 4: Likhetho tsa Lijo
A re re re na le lenane la dijo le nang le dikgetho tse 10 tsa dijo, mme re batla ho kgetha dijo tse 4 bakeng sa mokete. Re ka sebedisa metswako ho fumana palo ya ditsela tsa ho kgetha dijo tse 4 ho tse 10.
\[ \binom{10}{4} = 210 \]
Kahoo, ho na le litsela tse 210 tsa ho khetha lijo tse 4 ho tsoa likhethong tse 10.
Qetello
Metswako ke mohopolo oa bohlokoa lipalo 'me e na le lits'ebeliso tse pharaletseng mafapheng a fapaneng a mahlale le bophelong ba letsatsi le letsatsi. Ka ho utloisisa mohopolo oa metswako le mokhoa oa ho e bala, re ka rarolla mathata a fapaneng habonolo a amanang le ho khetha lintho ntle le ho nahana ka tatellano. Metswako e thusa ho baleng menyetla, tlhahlobo ea lipalo-palo, khopolo-taba ea linomoro, esita le ho hlophiseng mesebetsi ea letsatsi le letsatsi. Re tšepa hore sengoloa sena se fane ka kutloisiso e betere ea metswako le thuso ea eona maemong a fapaneng.
