Laʻana o nā Nīnau Kūkākūkā Hui Pū ʻIa
He manaʻo koʻikoʻi nā hui pū ʻana i ke kumumanaʻo probabilidad a me nā helu helu, e hiki ai iā mākou ke helu i ka helu o nā ala e koho ai i nā mea mai kahi set me ka nānā ʻole i ke kauoha. I nā hui pū ʻana, ʻaʻole loli ke kauoha o ke koho ʻana, ʻaʻole like me nā permutations, kahi e koʻikoʻi ai ke kauoha o ke koho ʻana. Hoʻohana ʻia nā hui pū ʻana ma nā ʻano kahua like ʻole, mai ka ʻepekema kamepiula a me ka biology a hiki i ka hoʻokele waiwai a me ka makemakika. E wehewehe kēia ʻatikala i ke kumumanaʻo kumu o nā hui pū ʻana a hāʻawi i kekahi mau pilikia hoʻohālike a me nā kūkākūkā e kōkua i kou hoʻomaopopo ʻana.
Manaʻo Kumu o ka Hoʻohuihui ʻana
Ma mua o ko kākou neʻe ʻana i nā pilikia hoʻohālike, e hoʻomaka kākou me kahi ʻike maʻamau o nā hui pū ʻana.
ʻO ka hui pū ʻana o nā mea \(n\) i lawe ʻia \(r\) i ka manawa hoʻokahi e hōʻike ʻia e ka hōʻailona \(\binom{n}{r}\) a i ʻole \(\mathbf{C}(n, r)\). ʻO ke ʻano hana no ka helu ʻana i ka hui pū ʻana penei:
\[ \binom{n}{r} = \frac{n!}{r!(nr)!} \]
– ʻO \( n \) ka huina o nā mea i loaʻa.
– ʻO \( r \) ka helu o nā mea e koho ʻia.
– ! (factorial) he hana makemakika e hoʻonui ai i kahi helu me nā helu maikaʻi a pau ma lalo a hiki i ka 1.
Nā Nīnau Laʻana a me ke Kūkākūkā
Laʻana Nīnau 1: Ke koho ʻana i kahi hui mai kahi hui o nā haumāna
Nīnau:
He 10 mau haumāna i loko o kahi papa. Ehia mau ala e hiki ai ke koho ʻia he 4 mau haumāna mai ka papa e komo i ka hoʻokūkū?
Kūkākūkā:
Ke hoʻohana nei i ke ʻano hui pū ʻana:
\[ \binom{10}{4} = \frac{10!}{4! \cdot (10-4)!} = \frac{10!}{4! \cdot 6!} \]
Ke helu nei i ka factorial:
– \( 10! = 10 \manawa 9 \manawa 8 \manawa 7 \manawa 6! \)
– \( 4! = 4 \manawa 3 \manawa 2 \manawa 1 = 24 \)
– Aia mua ʻo \( 6! \) ma luna, no laila hiki iā ia ke hoʻopau i ka 6! ma \(10!\).
No laila:
\[ \binom{10}{4} = \frac{10 \times 9 \times 8 \times 7}{4 \times 3 \times 2 \times 1} = \frac{5040}{24} = 210 \]
No laila, aia he 210 mau ala e koho ai i 4 mau haumāna mai 10 mau haumāna.
Laʻana Nīnau 2: Nā Hui Pū ʻana me ka Hoʻohana ʻana i nā Kāleka
Nīnau:
Mai kahi papa kāleka he 52, ehia mau ala e koho ai i kahi lima poker 5-kāleka?
Kūkākūkā:
Ke hoʻohana nei i ke ʻano hui pū ʻana:
\[ \binom{52}{5} = \frac{52!}{5! \cdot (52-5)!} = \frac{52!}{5! \cdot 47!} \]
Ke helu nei i ka factorial:
– \( 52! = 52 \manawa 51 \manawa 50 \manawa 49 \manawa 48 \manawa 47! \)
– \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \)
– Aia mua ʻo \( 47! \) ma luna, no laila hiki iā ia ke hoʻopau i ka 47! ma \(52!\).
No laila:
\[ \binom{52}{5} = \frac{52 \times 51 \times 50 \times 49 \times 48}{5 \times 4 \times 3 \times 2 \times 1} = \frac{311875200}{120} = 2598960 \]
No laila, aia he 2,598,960 mau ala e koho ai i 5 kāleka mai 52 mau kāleka.
Laʻana Nīnau 3: Ke helu ʻana i nā hui pū ʻana no ke koho ʻana i nā hui haʻuki
Nīnau:
He 15 mau mea pāʻani o kahi hui kinipōpō hīnaʻi. Inā makemake ke kumu aʻo e koho i 5 mau mea pāʻani ma ke ʻano he poʻe hoʻomaka, ehia mau ala e hiki ai ke hana ʻia kēia?
Kūkākūkā:
Ke hoʻohana nei i ke ʻano hui pū ʻana:
\[ \binom{15}{5} = \frac{15!}{5! \cdot (15-5)!} = \frac{15!}{5! \cdot 10!} \]
Ke helu nei i ka factorial:
– \( 15! = 15 \manawa 14 \manawa 13 \manawa 12 \manawa 11 \manawa 10! \)
– \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \)
– Aia mua ʻo \( 10! \) ma luna, no laila hiki iā ia ke hoʻopau i ka 10! ma \(15!\).
No laila:
\[ \binom{15}{5} = \frac{15 \times 14 \times 13 \times 12 \times 11}{5 \times 4 \times 3 \times 2 \times 1} = \frac{360360}{120} = 3003 \]
No laila, aia he 3,003 mau ala e koho ai i 5 mau mea pāʻani mai 15 mau mea pāʻani.
Laʻana Nīnau 4: Nā Hui Pū ʻana ma ka Biology
Nīnau:
I loko o kahi māla, aia he 8 ʻano pua like ʻole. Ehia mau ala e hiki ai ke koho ʻia nā ʻano pua 3 e hana i kahi hoʻonohonoho pua?
Kūkākūkā:
Ke hoʻohana nei i ke ʻano hui pū ʻana:
\[ \binom{8}{3} = \frac{8!}{3! \cdot (8-3)!} = \frac{8!}{3! \cdot 5!} \]
Ke helu nei i ka factorial:
– \( 8! = 8 \manawa 7 \manawa 6 \manawa 5! \)
– \( 3! = 3 \manawa 2 \manawa 1 = 6 \)
– Aia mua ʻo \( 5! \) ma luna, no laila hiki iā ia ke hoʻopau i ka 5! ma \(8!\).
No laila:
\[ \binom{8}{3} = \frac{8 \times 7 \times 6}{3 \times 2 \times 1} = \frac{336}{6} = 56 \]
No laila, aia he 56 mau ala e koho ai i 3 ʻano pua mai 8 ʻano pua.
Laʻana Nīnau 5: Hoʻohuihui ʻana i ke Kūkulu Hui
Nīnau:
He 12 mau puke like ʻole kāu ma kahi papa puke. Ehia mau ala e hiki ai iā ʻoe ke koho i 5 mau puke mai nā 12?
Kūkākūkā:
Ke hoʻohana nei i ke ʻano hui pū ʻana:
\[ \binom{12}{5} = \frac{12!}{5! \cdot (12-5)!} = \frac{12!}{5! \cdot 7!} \]
Ke helu nei i ka factorial:
– \( 12! = 12 \manawa 11 \manawa 10 \manawa 9 \manawa 8 \manawa 7! \)
– \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \)
– Aia mua ʻo \( 7! \) ma luna, no laila hiki iā ia ke hoʻopau i ka 7! ma \(12!\).
No laila:
\[ \binom{12}{5} = \frac{12 \times 11 \times 10 \times 9 \times 8}{5 \times 4 \times 3 \times 2 \times 1} = \frac{95040}{120} = 792 \]
No laila, aia he 792 mau ala e koho ai i 5 mau puke mai loko mai o 12 mau puke.
Ka hopena
He manaʻo pono loa nā hui ʻana ma nā ʻano like ʻole, ʻoiai ke pono mākou e ʻike i ka helu o nā ala e koho ai i nā mea me ka nānā ʻole i ke kauoha. Ma ka hoʻohana ʻana i ke ʻano hui kumu \(\binom{n}{r}\), hiki iā mākou ke helu wikiwiki a maʻalahi iā lākou, inā maopopo iā mākou pehea e hana ai nā factorials. Ma o nā hiʻohiʻona ma luna, ke manaʻolana nei mākou e loaʻa kahi ʻike piha a maopopo o nā hui ʻana. E hoʻomau i ka hoʻomaʻamaʻa ʻana me nā pilikia like ʻole e hoʻoikaika hou i kou ʻike.
