Ngā Ture mō te Whakakī i ngā Wāhi i roto i te Pāngarau
Ko ngā ture whakakī-ātea, e mōhiotia ana ko ngā ture whakarōpū me ngā ture whakakotahi, he ariā taketake i roto i te tūponotanga me te tatauranga. Mā ēnei ture ka taea e tātou te tatau i te maha o ngā huarahi rerekē hei whakarite, hei tīpako rānei i tētahi kohinga o ngā mea. I roto i tēnei tuhinga, ka tūhuratia e tātou ngā ariā taketake, ngā tono, me ngā tauira o te ao tūturu o ngā ture whakakī-ātea.
Te Māramatanga Taketake
I roto i te pāngarau, ka whakamahia ngā ture whakakī-wāhi hei tatau i te maha o ngā huarahi rerekē hei whakarite, hei tīpako rānei i ngā huānga i roto i tētahi huinga. E rua ngā ariā matua o ēnei ture: ngā whakarerekētanga me ngā huinga.
Te whakarerekētanga
Ko te whakarerekētanga he whakaritenga anō o ngā mea i roto i tētahi raupapatanga motuhake. I roto i ngā whakarerekētanga, he mea tino nui te raupapatanga. Hei tauira, ko te whakarerekētanga o ngā mea e toru A, B, me C ko:
– ABC
– ACB
– BAC
– BCA
– Teihana Whakahaere
– CBA
Mena he n ā tātou mea, ko te maha o ngā whakarerekētanga o ngā mea n ko n!. Ko te tohu tauwehe (n!) te tikanga o te whakareatanga o ngā tauoti pai katoa tae atu ki te n. Hei tauira, 3! = 3 × 2 × 1 = 6.
Ki te hiahia tātou ki te tatau i ngā whakarerekētanga o ngā mea n i tangohia ko r i te wā kotahi, ka whakamahia e tātou te tātai whakarerekētanga:
\[ P(n, r) = \frac{n!}{(nr)!} \]
Huinga
Ko te huinga ko te kōwhiringa o ngā mea me te kore e aro ki te raupapa. Hei tauira, ko te huinga o ngā mea e toru, arā, a A, a B, me C, e rua ngā mea e tangohia ana i te wā kotahi:
– AB
– AC
– BC
Ko te maha o ngā huinga o ngā mea n i tangohia ko r i te wā kotahi ka tohua e \( C(n, r) \) me \( \binom{n}{r} \), ā, ka tatauhia mā te tātai:
\[ C(n, r) = \frac{n!}{r!(nr)!} \]
Te Whakatinanatanga o ngā Ture Whakakī Wāhi
He maha ngā whakamahinga whai hua o ngā ture whakakī-ātea i roto i ngā mara pērā i te tatauranga, te tūponotanga, te pūtaiao rorohiko, me te rangahau pūtaiao.
I roto i ngā Tatauranga
I roto i ngā tatauranga, ka whakamahia ngā ture whakakī-ātea hei tatau i te maha o ngā huarahi ka taea te whakarite raraunga. Hei tauira, i roto i tētahi rangahau, ka hiahia pea tātou ki te mōhio e hia ngā huarahi ka taea e tātou te whiriwhiri i tētahi tauira mai i tētahi taupori.
I roto i te Tūponotanga
I roto i te tūponotanga, ka āwhina ngā ture whakakī-tūnga ki te tatau i te tūponotanga o te puta o tētahi kaupapa. Hei tauira, ka taea e tātou te tatau i te tūponotanga o te whiwhi i tētahi huinga kāri i roto i tētahi kēmu pōka.
I roto i te Pūtaiao Rorohiko
I roto i te pūtaiao rorohiko, ka whakamahia ngā ture whakakī-wāhi i roto i ngā rauropi me ngā hanganga raraunga. Hei tauira, i roto i te hōtaka, ka hiahia pea tātou ki te mōhio ki te maha o ngā huarahi rerekē hei whakarōpū raraunga.
Ngā Pātai Tauira me te Kōrero
Hei mārama ake, me titiro tātou ki ētahi tauira pātai me ā rātou matapakinga.
Tauira 1: Te Whakarerekētanga Kore Whakahoki
E hia ngā huarahi ka taea e koe te whakarite i te kupu "PĀNGARAU"?
Tekau ngā reta o te kupu "MATHEMATICS", ko ētahi he mea tāruarua. Hei tatau i te maha o ngā whakarerekētanga o tēnei kupu, ka whakamahia e mātou te tātai:
\[ \frac{n!}{k_1! \cdot k_2! \cdot \ldots \cdot k_m!} \]
ko \( n \) te tapeke o ngā reta, ā, ko \( k_1, k_2, \ldots, k_m \) te maha o ngā tāruaruatanga o ia reta. I roto i te kupu “MATHEMATICS”:
– M: 2 ngā wā
– A: 3 ngā wā
– T: 2 ngā wā
– E: 1 wā
– Ahau: 1 wā
– K: 1 wā
Nō reira, ko te maha o ngā whakarerekētanga ko:
\[ \frac{10!}{2! \cdot 3! \cdot 2! \cdot 1! \cdot 1! \cdot 1!} = \frac{3628800}{2 \cdot 6 \cdot 2 \cdot 1 \cdot 1 \cdot 1} = \frac{3628800}{24} = 151200 \]
Nō reira, e 151200 ngā huarahi hei whakarite i te kupu "PĀNGARAU".
Tauira 2: Huinga
E hia ngā huarahi hei whiriwhiri i ngā ākonga e 3 mai i ngā ākonga e 5?
Ka whakamahia e mātou te tātai whakakotahi:
\[ C(n, r) = \frac{n!}{r!(nr)!} \]
Me n = 5 me r = 3:
\[ C(5, 3) = \frac{5!}{3!(5-3)!} = \frac{120}{6 \cdot 2} = \frac{120}{12} = 10 \]
Nō reira, e 10 ngā huarahi hei whiriwhiri i ngā ākonga e 3 mai i ngā ākonga e 5.
Tauira 3: Te Whakarerekētanga me te Whakahokinga
E hia ngā huarahi ka taea e koe te whakarite i te kupu "POUROU" mēnā ka puta rua te reta O?
E rima ngā reta o te kupu "BALLOON" me te reta kotahi e whakahuatia ana (O). Ka whakamahia e mātou te tātai:
\[ \frac{n!}{k!} \]
ko n te tapeke o ngā reta, ā, ko k te maha o ngā tāruaruatanga o ngā reta. I roto i te kupu “BALLOON”:
– n = 5
– k = 2 (reta O)
Nō reira, ko te maha o ngā whakarerekētanga ko:
\[ \frac{5!}{2!} = \frac{120}{2} = 60 \]
Nō reira, e 60 ngā huarahi hei whakarite i te kupu "PŌUROU" me te reta O e puta rua ana.
Whakamutunga
He ariā nui ngā ture whakakī-wāhi i roto i te pāngarau e whakamahia ana hei tatau i te maha o ngā huarahi rerekē hei whakarite, hei tīpako rānei i ngā huānga i roto i tētahi huinga. Mā te mārama ki ngā whakarerekētanga me ngā huinga ka taea e tātou te whakaoti rapanga maha i roto i te tūponotanga, te tatauranga, me te maha atu o ngā mara. Mā te mārama me te matatau ki ēnei ariā ka huaki ngā whai wāhitanga maha mō te tātari me te whakaoti rapanga uaua ake i roto i ngā momo marautanga.
