hui ʻana
Pendahuluan
I ka makemakika, ʻo ka hui pū ʻana he manaʻo nui ia i hoʻohana ʻia e hoʻoholo i ka helu o nā ala e koho ai i kahi hui o nā mea me ka nānā ʻole i ko lākou hoʻonohonoho ʻana. Hoʻohana pinepine ʻia nā hui pū ʻana ma nā ʻano ʻepekema like ʻole, me nā helu helu, ka hiki, a me ka helu ʻana, a me ke ola o kēlā me kēia lā, e like me ka hoʻonohonoho ʻana, nā pāʻani kāleka, a me nā mea hou aku. E ʻimi hohonu kēia ʻatikala i ke kumumanaʻo o nā hui pū ʻana, nā haʻilula i hoʻohana ʻia, a me kekahi o kā lākou mau noi pono.
Ka Wehewehena o ka Hui Pū ʻana
ʻO ka hui pū ʻana he ala ia e koho ai i kahi helu o nā mea mai kahi set me ka nānā ʻole i ke kaʻina o ko lākou koho ʻia ʻana. Inā he \( n \) mau mea like ʻole a makemake mākou e koho \( r \) mau mea mai ka set, a laila hiki ke hōʻike ʻia ka hui pū ʻana e ka hōʻailona \( C(n, r) \) a i ʻole \( \binom{n}{r} \).
Formula Hoʻohuihui
ʻO ke ʻano hana no ka helu ʻana i ka helu o nā hui pū ʻana \( \binom{n}{r} \) penei:
\[ \binom{n}{r} = \frac{n!}{r! (nr)!} \]
Ma kahi o \( n! \) (heluhelu ʻia "n factorial") ka huahana o nā helu helu maikaʻi a pau a hiki i \( n \). No ka laʻana, \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \).
Laʻana Helu Hui Pū ʻIa
Manaʻo mākou he 5 mau mea like ʻole, a makemake mākou e koho i 3 mau mea mai ka set. Ehia mau ala e hiki ai iā mākou ke hana i kēia?
Ke hoʻohana nei i ke ʻano hui pū ʻana:
\[ \binom{5}{3} = \frac{5!}{3! (5-3)!} = \frac{5!}{3! \cdot 2!} = \frac{120}{6 \cdot 2} = 10 \]
No laila, aia he 10 mau ala e koho ai i 3 mau mea mai 5 mau mea me ka hoʻolohe ʻole i ke kaʻina.
Nā Hui Pū ʻana ma ka Pahiki
Hoʻohana pinepine ʻia nā hui pū ʻana i ka probability e hoʻoholo ai i ka probability o kahi hanana. No ka laʻana, i nā pāʻani kāleka, hiki iā mākou ke hoʻohana i nā hui pū ʻana e helu i ka probability o ka loaʻa ʻana o kahi lima.
Laʻana 1: Pāʻani Kāleka
Manaʻo mākou makemake mākou e helu i ka hiki ke loaʻa i kahi pālua o nā Aces i loko o ʻelima mau kāleka i hāʻawi ʻia mai kahi papa maʻamau o 52 mau kāleka. Hiki iā mākou ke hoʻohana i nā hui pū ʻana e helu i ka huina o nā ala e koho ai i ʻelima mau kāleka mai 52 mau kāleka a me ka helu o nā ala e koho ai i ʻelua Aces mai ʻehā Aces.
Nā ala āpau e koho ai i 5 kāleka mai 52:
\[ \binom{52}{5} = \frac{52!}{5! \cdot 47!} \]
Pehea e koho ai i 2 Aces mai 4 Aces:
\[ \binom{4}{2} = \frac{4!}{2! \cdot 2!} = 6 \]
Pehea e koho ai i 3 kāleka mai nā kāleka 48 i koe (ʻoiai ua koho mua ʻia 2 kāleka mai nā 4 Aces):
\[ \binom{48}{3} = \frac{48!}{3! \cdot 45!} \]
No laila, ʻo ka likelika o ka loaʻa ʻana o ʻelua Aces ma kahi lima ʻelima kāleka:
\[ P = \frac{\binom{4}{2} \cdot \binom{48}{3}}{\binom{52}{5}} \]
Nā Hui Pū ʻana ma nā Heluhelu
I loko o nā helu helu, hoʻohana ʻia nā hui ʻana i nā ʻano loiloi like ʻole, me ka laʻana a me ka hoʻāʻo kuhiakau. Kōkua nā hui ʻana i ka hoʻoholo ʻana i ka helu o nā ala e koho ai i kahi laʻana kikoʻī mai kahi heluna kanaka.
Laʻana 2: Laʻana
Manaʻo mākou he 20 kānaka ko mākou heluna kanaka a makemake mākou e koho i kahi hāpana o 4 kānaka. Hiki iā mākou ke hoʻohana i nā hui pū ʻana e hoʻoholo ai i ka helu o nā ala e koho ai i kēlā hāpana.
\[ \binom{20}{4} = \frac{20!}{4! \cdot 16!} \]
Hāʻawi kēia helu i ka ʻike koʻikoʻi i nā ʻano loiloi helu like ʻole, e like me ka helu ʻana i ka hiki ke loaʻa a i ʻole ka kuhi ʻana o ka palena.
Hoʻohuihui a me ke Kumumanaʻo Helu
He kuleana koʻikoʻi nō hoʻi nā hui pū ʻana i ke kumumanaʻo helu, ʻoi aku hoʻi i loko o ka pōʻaiapili o nā coefficients binomial a me nā ʻike makemakika ʻē aʻe. No ka laʻana, hoʻohana ka Binomial Theorem i nā hui pū ʻana e hoʻomohala i nā hōʻike algebraic.
ʻO ke kumumanaʻo Binomial
ʻŌlelo ka Binomial Theorem:
\[ (x + y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{nk} y^k \]
Maanei, hōʻike ka helu o nā ala e koho ai i nā mea \( k \) mai nā mea \( n \).
Nā Hoʻopili Hoʻohuihui i ke Ola o kēlā me kēia lā
ʻAʻole i kaupalena wale ʻia ke kumumanaʻo o nā hui pū ʻana i ka makemakika kumumanaʻo, akā he nui nā noi pono i ke ola o kēlā me kēia lā.
Laʻana 3: Nā Hoʻonohonoho Papa Hana
Manaʻo mākou he 5 mau haʻawina e hoʻopau ai i hoʻokahi pule, a makemake mākou e koho i 3 e hana i ka Pōʻakahi. Hiki iā mākou ke hoʻohana i nā hui pū ʻana e hoʻoholo ai i ka helu o nā ala e koho ai i 3 mau haʻawina mai kēlā mau 5.
\[ \binom{5}{3} = 10 \]
No laila, aia he 10 mau ala e hoʻonohonoho ai i ka papa hana.
Laʻana 4: Nā Koho Meaʻai
Manaʻo mākou he papa kuhikuhi me 10 mau koho meaʻai, a makemake mākou e koho i 4 mau kīʻaha no kahi pāʻina. Hiki iā mākou ke hoʻohana i nā hui pū ʻana e hoʻoholo i ka helu o nā ala e koho ai i 4 mau kīʻaha mai 10.
\[ \binom{10}{4} = 210 \]
No laila, aia he 210 mau ala e koho ai i 4 mau meaʻai mai 10 mau koho.
Ka hopena
He manaʻo koʻikoʻi nā hui ʻana i ka makemakika a he nui nā noi ma nā ʻano ʻepekema like ʻole a me ke ola o kēlā me kēia lā. Ma ka hoʻomaopopo ʻana i ke kumumanaʻo o nā hui ʻana a pehea e helu ai iā lākou, hiki iā mākou ke hoʻoponopono maʻalahi i nā pilikia like ʻole e pili ana i ke koho ʻana i nā mea me ka nānā ʻole i ke kauoha. He kōkua nā hui ʻana i nā helu probabili, ka loiloi helu, ke kumumanaʻo helu, a hiki i ka hoʻonohonoho ʻana i nā hana o kēlā me kēia lā. Manaʻolana, ua hāʻawi kēia ʻatikala i kahi ʻike maikaʻi aʻe o nā hui ʻana a me ko lākou pono i nā ʻano like ʻole.
