Ngā Ture Whakarerekētanga me te Whakakotahitanga
I roto i te pāngarau, inā koa i ngā mara o te tūponotanga me te tatauranga, he maha ngā wā ka tūtaki tātou ki te pātai "e hia ngā huarahi" ka taea te puta o tētahi huihuinga. Hei tauira: e hia ngā whakaritenga nohoanga ka taea mō tētahi tokomaha o ngā tāngata? E hia ngā huarahi ka taea te whiriwhiri i ngā mema o te tīma mai i tētahi rōpū ākonga? Ka whakautua ngā pātai pēnei mā te whakamahi i ngā ture whakawhiti me te whakakotahitanga, arā, ngā ariā matua e rua i roto i ngā ture tatau. Ahakoa e pā ana ēnei ki "te maha o ngā huarahi", ko te rerekētanga nui kei te whakaarohia te raupapatanga.
1. Te Ariā Taketake o ngā Ture Tātaitanga
I mua i te ruku hohonu ki ngā whakarerekētanga me ngā whakakotahitanga, he ariā matua hei mārama: ko te tatau i te maha o ngā putanga ka taea mai i tētahi huinga tikanga kua hoatu. Ka taea te tatau ā-ringa mō ngā take iti, engari mō ngā take nui, me whai tātai whai hua.
Ko ngā mātāpono matua e rua mō te tatau ko:
1. Ture mō te Hua
Mena he maha ngā wāhanga o tētahi tukanga, ā, he maha ngā kōwhiringa o ia wāhanga, ko te tapeke o ngā huarahi te hua o te maha o ngā kōwhiringa i ia wāhanga.
2. Ture o te Tapeke
Mena ka taea te whakatau i tētahi kōwhiringa i roto i ētahi huarahi motuhake (kāore e taupatupatu), ko te tapeke o ngā huarahi ko te tapeke o aua huarahi.
Ko ngā whakarerekētanga me ngā whakakotahitanga he tono anō o tēnei mātāpono, inā koa ka tīmata te whakarite, te tīpako rānei i ngā mea.
2. Whakarerekētanga: Te Whakaritenga mā te Aro ki te Raupapa
Ko te whakarōpūtanga he huarahi whakarite, whiriwhiri rānei i ngā mea e tika ana te raupapa. Ko te tikanga he rerekē te whakaritenga AB i te BA.
a. Ngā whakarerekētanga o ngā mea rerekē e n (kua whakaritea katoa)
Mena e n ngā mea rerekē hei whakarite katoa i roto i tētahi raupapatanga, ko te maha o ngā whakaritenga ko:
\[
n! = n \times (n-1) \times (n-2) \times \dots \times 2 \times 1
\]
Ko te tohu "!" ka kiia ko te tohu whakarea.
Tauira:
E whā ngā pukapuka rerekē. E hia ngā huarahi ka taea te whakarite i ēnei ki runga i te whata?
\[
4! = 4 \whakareatia ki te 3 \whakareatia ki te 2 \whakareatia ki te 1 = 24
\]
Nō reira, e 24 ngā whakaritenga.
b. Te whakarerekētanga ā-wāhanga: te whiriwhiri i a r mai i a n (ka whakaarohia te raupapa)
Mena mai i ngā mea rerekē e n ka whiriwhiria e tātou ngā mea e r hei whakarite (ehara i te mea ko te katoa), ko te tātai whakatauira ko:
\[
P(n,r) = \frac{n!}{(nr)!}
\]
Tauira:
O ngā ākonga tokoono, e toru ngā ākonga ka whiriwhiria hei heamana, hei heamana tuarua, me hei hēkeretari. E hia ngā huarahi ka taea ai tēnei?
Nā te mea he rerekē ngā tūranga o te heamana-te tumuaki tuarua-hēkeretari, he mea nui te raupapatanga.
\[
P(6,3) = \frac{6!}{(6-3)!}=\frac{6!}{3!} = 6 \times 5 \times 4 = 120
\]
E 120 ngā huarahi.
c. Ngā whakarerekētanga me te mea kotahi (whakahokinga/ōrite)
I ētahi wā, kāore e motuhake katoa ngā mea. Hei tauira, i roto i te kupu "PŌ" e rua ngā M me ngā A e rua (mō "PŌ" rānei: e rua ngā M, e rua ngā A? Ina hoki, "PŌ" = PŌ: M=2, A=2, L=1). Ka tatauhia te maha o ngā whakaritenga rerekē mā te:
\[
\frac{n!}{n_1! \, n_2! \, \dots}
\]
ko \(n\) te tapeke o ngā mea, ā, ko \(n_1, n_2\) te maha o ngā mea ōrite.
Tauira:
E hia ngā whakaritenga rerekē o ngā reta i roto i te kupu “PŌ”?
Te maha o ngā reta \(n=5\), e 2 ngā reta a M, e 2 ngā reta a A, e 1 ngā reta a L.
\[
\frac{5!}{2!\,2!} = \frac{120}{4} = 30
\]
Nō reira, e 30 ngā whakaritenga rerekē.
3. Huinga: Kōwhiringa me te kore e aro ki te raupapatanga
Ko te whakakotahitanga he huarahi hei tīpako i ngā mea kāore he pānga o te raupapa. He rite tonu te tīpako i a A me B ki te tīpako i a B me A.
Ka tīpakohia e te tātai huinga a r mai i ngā mea n:
\[
C(n,r) = \binom{n}{r}=\frac{n!}{r!(nr)!}
\]
a. Tauira o tētahi huinga māmā
Tauira:
Mai i ngā ākonga 10, e 3 ngā ākonga ka whiriwhiria hei mema mō te tīma whakataetae (engari kāore he tūranga motuhake). E hia ngā huarahi?
Nā te mea kāore he tūnga, kāore te raupapa i te mea nui.
\[
C(10,3)=\frac{10!}{3!\,7!}=\frac{10 \times 9 \times 8}{3 \times 2 \times 1}=120
\]
E 120 ngā huarahi.
b. Te whanaungatanga i waenga i ngā whakarerekētanga me ngā huinga
Kia mōhio koe he hononga ngā whakarerekētanga me ngā huinga. Hei whiriwhiri i ngā tāngata r me te whakarite i a rātou, ka taea e tātou te:
– whiriwhiria tuatahitia ngā tāngata r: \(C(n,r)\)
– whakarite i te hunga r: \(r!\)
Nō reira:
\[
P(n,r) = C(n,r)\ngā wā r!
\]
E whakaatu ana tēnei he "nui ake" te whakarerekētanga nā te mea ka wehewehe i te raupapa.
4. Me pēhea te whakatau: Whakamahia te Whakarerekētanga, te Whakakotahitanga rānei?
Hei whakaoti rapanga, ko te taahiraa nui ko te mōhio mēnā kua whai whakaarohia te raupapa.
Whakamahia ngā whakarerekētanga mēnā:
– he tūranga, he taitara rānei (heamana, he tumuaki tuarua, he tūranga 1-2-3),
– he whakaritenga nohoanga kei reira,
– he waehere, he raupapa whakaritenga rānei.
Whakamahia he ranunga mēnā:
– ko ngā mema rōpū kua tohua anake,
– kāore te raupapa e wehewehe i te putanga,
– ko te mea nui ko wai ka pōtitia, ehara i te mea ko tōna tūranga.
Tauira tere:
– Kōwhiria kia 5 o roto i te 12 tāngata hei mema mō te komiti: he huinga
– Te whakatau i ngā toa tuatahi, tuarua, me tuatoru mai i te 12 o ngā kaiuru: te whakatauira
5. Ngā Tauira o ngā Whakamahinga i roto i te Oranga o Ia Rā
Kāore ngā whakapūtanga me ngā whakakotahitanga e kitea ana i roto i ngā pukapuka pāngarau anake, engari i roto anō hoki i ngā āhuatanga tūturu:
1. Haumarutanga kupuhipa (kupuhipa/PIN)
Ko te maha o ngā PIN mati-4 e taea ana (0–9) me te tukurua e whakaaetia ana ko \(10^4\). E pā ana tēnei ki te ture whakarea me te ariā o ngā whakarerekētanga me te tukurua.
2. Te whakarite i ngā wātaka, i ngā nohoanga rānei
Te whakatau i ngā tūranga nohoanga i ngā huihuinga ōkawa mā te whakamahi i ngā whakarerekētanga e pā ana ki ngā tūranga rerekē.
3. Te whiriwhiri i te tīma, i te komiti rānei
He whakakotahitanga te whiriwhiri i ētahi tāngata mai i tētahi rōpū, nā te mea kāore te raupapa i te mea nui.
4. Ngā kēmu kāri
He maha ngā wā ka whakamahia ngā huinga hei tatau i te tūponotanga o tētahi ringa motuhake i te pōka, i ētahi atu kēmu rānei.
6. Ngā Hapa Noa hei Ārai
Ko ētahi hapa e puta pinepine ana i te wā e mahi ana i ngā raruraru whakarōpū me te whakakotahitanga:
– Te whakaaro he iti noa te raupapa ahakoa he mea nui, hei tauira, te whiriwhiri i te heamana me te heamana tuarua (me whai i te whakarerekētanga).
– Te wareware ki te wehewehe i ngā mea ōrite, pērā i te tito kupu he reta tāruarua kei roto.
– Te tatau hē i ngā tauwehenga, inā koa i te whakangawari i te āhua \(\frac{n!}{(nr)!}\).
Ko tētahi huarahi hei ārai i tēnei ko te tuhi i te whakamārama o te pātai ki roto i ngā rerenga kōrero māmā: "Me whiriwhiri, me whakarite rānei ahau?" me "He rerekētanga te putanga mai i te tūranga?"
Te Katinga
He taputapu nui ngā ture o te whakarerekētanga me te whakakotahitanga mō te tatau i te maha o ngā mea ka taea i roto i ngā āhuatanga rerekē. Ka whakamahia ngā whakarerekētanga ina he mea nui te raupapa, te tūranga rānei, ko ngā whakakotahitanga ia ka whakamahia ina kāore he mea nui te raupapa. Mā te mārama ki tēnei rerekētanga, te matatau ki ngā tauwehenga, me te whakamahi i ngā tātai tika, ka taea e tātou te whakaoti tere me te tika i ngā raruraru tatau me ngā tūponotanga. I roto i te mahi, ko te kaha ki te whiriwhiri i te tikanga tika—te whakarerekētanga, te whakakotahitanga rānei—he mea nui ake i te maumahara noa i ngā tātai.
