Nā nīnau hoʻohālike kūkākūkā Combinatorics
He lālā o ka makemakika ʻo Combinatorics e aʻo ana i ka helu ʻana, ka hoʻonohonoho ʻana, a me nā ʻano hana o nā seti o nā mea. He nui nā noi o Combinatorics ma nā ʻano like ʻole, me ka ʻepekema kamepiula, nā helu helu, ka biology, a me ka hoʻokele waiwai. Ma kēia ʻatikala, e kūkākūkā mākou i kekahi mau laʻana a me kā lākou mau kūkākūkā e pili ana i ka combinatorics, kahi e manaʻolana ai e hāʻawi i kahi ʻike maikaʻi aʻe o nā manaʻo kumu a me nā noi o ka combinatorics.
Nīnau 1: Hoʻololi
Nīnau:
Ehia mau ala e hiki ai ke hoʻonohonoho ʻia nā puke like ʻole he 5 ma luna o kahi papa?
Kūkākūkā:
ʻO ka permutation ka hoʻonohonoho ʻana o nā mea i kahi hoʻonohonoho i kauoha ʻia. Ke koʻikoʻi ke kauoha, hoʻohana mākou i nā permutations. Ma ke ʻano o kēia pilikia, loaʻa iā mākou ʻelima mau puke like ʻole e hoʻonohonoho. ʻO ka helu o nā ala e hoʻonohonoho ai i kēia mau puke ʻelima:
\[ 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \]
No laila, aia he 120 mau ala e hoʻonohonoho ai i 5 mau puke like ʻole ma kahi papa.
Nīnau 2: Hoʻohuihui
Nīnau:
Mai loko mai o nā kānaka he 10, ehia mau ala e hana ai i kahi hui o 4 kānaka?
Kūkākūkā:
ʻO ka hui pū ʻana ke koho ʻana o nā mea kahi i koʻikoʻi ʻole ai ke kauoha. ʻO ke ʻano hana no ka hui pū ʻana:
\[ \binom{n}{k} = \frac{n!}{k!(nk)!} \]
Ma ke ʻano o kēia pilikia, \( n = 10 \) a me \( k = 4 \). No laila,
\[ \binom{10}{4} = \frac{10!}{4! \times (10-4)!} = \frac{10!}{4! \times 6!} \]
Ua ʻike mākou \( 10! = 10 \times 9 \times 8 \times 7 \times 6! \), a laila
\[ \binom{10}{4} = \frac{10 \times 9 \times 8 \times 7 \times 6!}{4! \times 6!} = \frac{10 \times 9 \times 8 \times 7}{4 \times 3 \times 2 \times 1} = 210 \]
No laila, aia he 210 mau ala e hana ai i kahi hui o 4 mai loko mai o 10 mau kānaka.
Nīnau 3: Nā Permutations me ka hana hou ʻana
Nīnau:
Ehia mau ala e hoʻonohonoho ai i ka huaʻōlelo "LEVEL"?
Kūkākūkā:
ʻO ka huaʻōlelo "LEVEL" he 5 mau leka, a ua hana hou ʻia kekahi o ia mau mea (L ʻelua a me E ʻelua). ʻO ke ʻano hoʻololi me ka hana hou ʻana penei:
\[ \frac{n!}{n_1! \times n_2! \times \ldots \times n_k!} \]
Ma ke ʻano o kēia pilikia, \( n = 5 \), \( n_1 = 2 \) no ka leka L, a me \( n_2 = 2 \) no ka leka E. No laila,
\[ \frac{5!}{2! \times 2!} = \frac{5 \times 4 \times 3 \times 2 \times 1}{2 \times 1 \times 2 \times 1} = \frac{120}{4} = 30 \]
No laila, aia he 30 mau ala e hoʻonohonoho ai i ka huaʻōlelo "LEVEL".
Nīnau 4: Hoʻohui pū ʻia me ka hana hou ʻana
Nīnau:
Ehia mau ala e koho ai i 3 mau lole mai loko mai o 5 mau ʻano lole like ʻole me ka ʻae ʻia ʻana o nā hana hou?
Kūkākūkā:
Hoʻohui pū ʻia me ka hana hou ʻana me ka hoʻohana ʻana i kēia ʻano hana:
\[ \binom{n+r-1}{r} \]
Ma ke ʻano o kēia pilikia, \( n = 5 \) (nā ʻano o nā mea ʻono) a me \( r = 3 \) (ka helu o nā mea ʻono i koho ʻia). No laila,
\[ \binom{5+3-1}{3} = \binom{7}{3} = \frac{7!}{3! \times 4!} \]
Ke ʻike nei \( 7! = 7 \times 6 \times 5 \times 4! \), a laila
\[ \binom{7}{3} = \frac{7 \times 6 \times 5 \times 4!}{3! \times 4!} = \frac{7 \times 6 \times 5}{3 \times 2 \times 1} = 35 \]
No laila, aia he 35 mau ala e koho ai i 3 mau lole mai 5 mau ʻano lole like ʻole me ka ʻae ʻia ʻana o nā hana hou.
Nīnau 5: Ke Kumumanaʻo o ka Hoʻohui
Nīnau:
Ehia mau ala e koho ai i hoʻokahi hua mai loko mai o kahi hīnaʻi me 3 ʻāpala, 2 ʻalani, a me 5 maiʻa?
Kūkākūkā:
ʻŌlelo ke kumumanaʻo o ka hoʻohui inā he nui nā ala e hana ai i kahi hana, a laila ʻo ka huina o nā ala ka huina o kēlā mau ala āpau. Ma ke ʻano o kēia pilikia,
– Aia he 3 mau ala e koho ai i hoʻokahi ʻāpala.
– ʻElua ala e koho ai i hoʻokahi ʻalani.
– Aia he 5 mau ala e koho ai i hoʻokahi maiʻa.
Nā ala āpau:
\[ 3 + 2 + 5 = 10 \]
No laila, aia he 10 mau ala e koho ai i hoʻokahi hua mai ka hīnaʻi.
Nīnau 6: Ke Kumumanaʻo o ka Hoʻonui ʻana
Nīnau:
Ehia mau ala e koho ai i hoʻokahi pālule mai nā koho 4 a me hoʻokahi pālule mai nā koho 3?
Kūkākūkā:
ʻŌlelo ke kumumanaʻo hoʻonui inā he nui nā ala e hana ai i ka hana mua a me nā ala he nui e hana ai i ka hana ʻelua, a laila ʻo ka huina o nā ala e hana ai i nā hana ʻelua ka hopena o nā ala e hana ai i kēlā me kēia hana.
Ma ke ʻano o kēia nīnau,
– Aia he 4 mau ala e koho ai i hoʻokahi pālule.
– Aia he 3 mau ala e koho ai i hoʻokahi pālule.
Nā ala āpau:
\[ 4 \times 3 = 12 \]
No laila, aia he 12 mau ala e koho ai i hoʻokahi pālule a me hoʻokahi pālule.
Ka hopena
ʻO Combinatorics, ma ke ʻano he lālā o ka makemakika, hāʻawi i kahi ʻano waiwai o nā ʻano hana a me nā manaʻo no ka helu ʻana a me ka hoʻonohonoho ʻana i nā mea like ʻole. Mai nā permutations a me nā hui pū ʻana a hiki i nā kumumanaʻo o ka hoʻohui a me ka hoʻonui ʻana, hoʻohana pinepine ʻia kēia mau manaʻo i nā ʻano hana like ʻole. Ma ka hoʻomaopopo ʻana i nā laʻana a me nā kūkākūkā ma luna, manaʻolana ʻia e hiki i ka poʻe heluhelu ke hoʻopili i nā manaʻo combinatorics i nā kūlana paʻakikī a hoʻomaikaʻi i kā lākou mau mākau hoʻoponopono pilikia ma ka makemakika a me nā aʻo ʻē aʻe.
