Nā Rula no ka Hoʻopiha ʻana i nā Wahi ma ka Makemakika
ʻO nā lula hoʻopiha hakahaka, i ʻike ʻia hoʻi he mau lula permutation a me ka hui pū ʻana, he mau manaʻo nui ia i ka probability a me nā helu helu. ʻAe kēia mau lula iā mākou e helu i ka helu o nā ʻano like ʻole e hoʻonohonoho ai a koho paha i kahi hōʻiliʻili o nā mea. Ma kēia ʻatikala, e ʻimi mākou i nā manaʻo kumu, nā noi, a me nā hiʻohiʻona honua maoli o nā lula hoʻopiha hakahaka.
Ka Hoʻomaopopo Kumu
I ka makemakika, hoʻohana ʻia nā lula hoʻopiha wahi e helu ai i ka helu o nā ʻano like ʻole e hoʻonohonoho ai a koho paha i nā mea i loko o kahi set. Aia ʻelua mau manaʻo nui i loko o kēia mau lula: nā permutations a me nā hui pū ʻana.
Hoʻololi
ʻO ka permutation kahi hoʻonohonoho hou ʻana o nā mea ma kahi hoʻonohonoho kikoʻī. I nā permutations, he mea nui loa ke kauoha. No ka laʻana, ʻo ka permutation o ʻekolu mau mea A, B, a me C penei:
–ABC
– ACB
– BAC
– BCA
– CAB
– CBA
Inā loaʻa iā kākou nā mea n, ʻo ka helu o nā permutations o nā mea n he n!. ʻO ke kau inoa factorial (n!) ʻo ia hoʻi ka hoʻonui ʻia ʻana o nā helu helu maikaʻi a pau a hiki i ka n. No ka laʻana, 3! = 3 × 2 × 1 = 6.
Inā makemake mākou e helu i nā permutations o n mau mea i lawe ʻia iā r i ka manawa hoʻokahi, hoʻohana mākou i ke ʻano permutation:
\[ P(n, r) = \frac{n!}{(nr)!} \]
hui ʻana
ʻO ka hui pū ʻana, ʻo ia ke koho ʻana o nā mea me ka nānā ʻole i ke kauoha. Eia kekahi laʻana, ʻo ka hui pū ʻana o ʻekolu mau mea A, B, a me C i lawe ʻia ʻelua i ka manawa hoʻokahi:
– AB
- AC
– BC
ʻO ka helu o nā hui pū ʻana o nā mea n i lawe ʻia ʻo r i ka manawa hoʻokahi ua hōʻike ʻia e \( C(n, r) \) a i ʻole \( \binom{n}{r} \), a ua helu ʻia e ke ʻano hana:
\[ C(n, r) = \frac{n!}{r!(nr)!} \]
Hoʻokō ʻana i nā Rula Hoʻopiha Wahi
He nui nā hoʻohana pono o nā lula hoʻopiha hakahaka ma nā kahua e like me ka helu helu, ka hiki, ka ʻepekema kamepiula, a me ka noiʻi ʻepekema.
Ma nā Heluhelu
I loko o nā helu helu, hoʻohana ʻia nā lula hoʻopiha hakahaka e helu i ka helu o nā ala hiki ke hoʻonohonoho i ka ʻikepili. No ka laʻana, i loko o kahi anamanaʻo, makemake paha mākou e ʻike i ka nui o nā ala e hiki ai iā mākou ke koho i kahi laʻana mai kahi heluna kanaka.
Ma ke Kūpono
I ka hiki ke hana ʻia, kōkua nā lula hoʻopiha wahi i ka helu ʻana i ka hiki ke hana ʻia kahi hanana. No ka laʻana, hiki iā mākou ke helu i ka hiki ke loaʻa kahi hui pū ʻana o nā kāleka i kahi pāʻani poker.
Ma ka ʻEpekema Kamepiula
I ka ʻepekema kamepiula, hoʻohana ʻia nā lula hoʻopiha wahi i nā algorithms a me nā ʻōnaehana ʻikepili. No ka laʻana, i ka papahana, makemake paha mākou e ʻike i ka helu o nā ʻano like ʻole e hoʻokaʻawale ai i ka ʻikepili.
Nā Nīnau Laʻana a me ke Kūkākūkā
I mea e maopopo hou aku ai, e nānā kākou i kekahi mau nīnau hoʻohālike a me kā lākou mau kūkākūkā ʻana.
Laʻana 1: Permutation me ka ʻole o ka hana hou ʻana
Ehia mau ala e hiki ai iā ʻoe ke hoʻonohonoho i ka huaʻōlelo "MATHEMATICS"?
ʻO ka huaʻōlelo "MATHEMATICS" he 10 mau leka, a ua hana hou ʻia kekahi o ia mau mea. No ka helu ʻana i ka helu o nā permutations o kēia huaʻōlelo, hoʻohana mākou i ke ʻano hana:
\[ \frac{n!}{k_1! \cdot k_2! \cdot \ldots \cdot k_m!} \]
kahi ʻo \( n \) ka huina o nā leka a ʻo \( k_1, k_2, \ldots, k_m \) ka helu o nā hana hou ʻana o kēlā me kēia leka. Ma ka huaʻōlelo "MATHEMATICS":
– M: 2 manawa
– A: 3 manawa
– T: 2 manawa
– E: 1 manawa
– ʻAu: 1 manawa
– K: 1 manawa
No laila, ʻo ka helu o nā permutations:
\[ \frac{10!}{2! \cdot 3! \cdot 2! \cdot 1! \cdot 1! \cdot 1!} = \frac{3628800}{2 \cdot 6 \cdot 2 \cdot 1 \cdot 1 \cdot 1} = \frac{3628800}{24} = 151200 \]
No laila, aia he 151200 mau ala e hoʻonohonoho ai i ka huaʻōlelo "MATHEMATICS".
Laʻana 2: Hoʻohuihui
Ehia mau ala e koho ai i 3 mau haumāna mai loko mai o 5 mau haumāna?
Hoʻohana mākou i ke ʻano hui pū ʻana:
\[ C(n, r) = \frac{n!}{r!(nr)!} \]
Me n = 5 a me r = 3:
\[ C(5, 3) = \frac{5!}{3!(5-3)!} = \frac{120}{6 \cdot 2} = \frac{120}{12} = 10 \]
No laila, aia he 10 mau ala e koho ai i 3 mau haumāna mai 5 mau haumāna.
Laʻana 3: Permutation me ka hana hou ʻana
Ehia mau ala e hiki ai iā ʻoe ke hoʻonohonoho i ka huaʻōlelo "BALLOON" inā ʻike ʻia ka leka O i ʻelua manawa?
ʻO ka huaʻōlelo "BALLOON" he 5 mau leka me hoʻokahi leka e hana hou ʻia ana (O). Hoʻohana mākou i ke ʻano hana:
\[ \frac{n!}{k!} \]
kahi ʻo n ka huina o nā leka a ʻo k ka helu o nā hana hou ʻana o nā leka. Ma ka huaʻōlelo "BALLOON":
– n = 5
– k = 2 (leka O)
No laila, ʻo ka helu o nā permutations:
\[ \frac{5!}{2!} = \frac{120}{2} = 60 \]
No laila, aia he 60 mau ala e hoʻonohonoho ai i ka huaʻōlelo "BALLOON" me ka leka O e ʻike ʻia ana i ʻelua manawa.
Ka hopena
He manaʻo koʻikoʻi nā lula hoʻopiha wahi i ka makemakika i hoʻohana ʻia e helu i ka helu o nā ʻano like ʻole e hoʻonohonoho ai a koho paha i nā mea i loko o kahi set. ʻO ka hoʻomaopopo ʻana i nā permutations a me nā hui pū ʻana e hiki ai iā mākou ke hoʻoponopono i nā pilikia like ʻole ma ka probability, statistics, a me nā ʻoihana ʻē aʻe he nui. ʻO ka hoʻomaopopo ʻana a me ka hoʻokele ʻana i kēia mau manaʻo e wehe i nā manawa kūpono he nui no ka nānā ʻana a me ka hoʻoponopono ʻana i nā pilikia paʻakikī ma nā ʻano aʻo like ʻole.
