Tauira o ngā Pātai Kōrero mō te Whakarerekētanga
Ko te whakarerekētanga ko te whakaritenga anō o tētahi huinga, o tētahi mea rānei i roto i tētahi raupapatanga motuhake. I roto i te pāngarau, e whakamahia whānuitia ana tēnei ariā hei tatau i te maha o ngā huarahi e taea ai te whakarite i tētahi rōpū mea. Kei raro nei, ka matapakihia e mātou ētahi tauira o ngā raruraru whakarerekētanga me ō rātou whakamārama whānui.
Te Whakamāramatanga o te Whakawhitiwhiti
Ko te whakarerekētanga o tētahi huinga he whakarerekētanga anō o ōna huānga i roto i tētahi raupapa motuhake. Mena he \( n \) ngā mea, ka tohua te whakarerekētanga e \( P(n) \) me te tika ake, \( P(n, r) \) mō ngā \( r \) whakarerekētanga o ngā \( n \) mea. Ko te tātai taketake mō te whakarerekētanga ko:
\[ P(n) = n! \]
ko te \( n! \) (n tauwehe) te hua o ngā tauoti pai katoa he iti iho i te, he ōrite rānei ki te \( n \).
I taua wā, ko te tātai whakarerekētanga \( r \) o ngā mea \( n \) ko:
\[ P(n, r) = \frac{n!}{(nr)!} \]
Ngā Pātai Tauira me te Kōrero
Tauira Pātai 1
He raruraru:
E hia ngā huarahi ka taea te whakarite i ngā pukapuka rerekē e whā ki runga i te whata?
Kōrero:
Hei whakarite i ngā pukapuka rerekē e whā, ka taea e tātou te whakamahi i te tātai whakarōpū hei tatau i ngā whakaritenga katoa o ngā pukapuka:
\[ P(4) = 4! = 4 \whakarea 3 \whakarea 2 \whakarea 1 = 24 \]
Nō reira, e 24 ngā huarahi hei whakarite i ngā pukapuka rerekē e 4 ki runga i te whata.
Tauira Pātai 2
He raruraru:
E hia ngā huarahi ka taea te whiriwhiri me te whakarite i ngā mema e 3 o tētahi tīma e 5 ngā mema i roto i tētahi raupapa kua hoatu?
Kōrero:
Ka whakamahia e mātou te tātai whakarerekētanga \( P(n, r) \) kei reira \( n = 5 \) me \( r = 3 \):
\[ P(5, 3) = \frac{5!}{(5-3)!} = \frac{5!}{2!} = \frac{5 \times 4 \times 3 \times 2!}{2!} = 5 \times 4 \times 3 = 60 \]
Nō reira, e 60 ngā huarahi hei tīpako me te whakarite i ngā mema e 3 o tētahi tīma e 5 ngā mema ki tētahi raupapa kua whakaritea.
Tauira Pātai 3
He raruraru:
E hia ngā huarahi e taea ai te whakarite i te kupu "PĀNGARAU" kia kore ai e tāruatia ngā reta?
Kōrero:
E whā ngā reta rerekē o te kupu "MATH". Ka taea e tātou te whakamahi i te tātai whakarōpū hei tatau i ngā whakaritenga katoa o ēnei reta:
\[ P(4) = 4! = 4 \whakarea 3 \whakarea 2 \whakarea 1 = 24 \]
Nō reira, e 24 ngā huarahi hei whakarite i ngā reta i roto i te kupu "MATH".
Tauira Pātai 4
He raruraru:
Mai i ngā tau 1, 2, 3, 4, 5, e hia ngā tau mati-toru ka taea te hanga mēnā kāore he mati e tāruatia ana?
Kōrero:
Hei hanga i tētahi tau mati-toru mai i ngā mati rerekē e 5, kāore he mati e tāruatia ana, ka whakamahia e mātou te whakarerekētanga \( P(5, 3) \):
\[ P(5, 3) = \frac{5!}{(5-3)!} = \frac{5!}{2!} = \frac{5 \times 4 \times 3 \times 2!}{2!} = 5 \times 4 \times 3 = 60 \]
Nō reira, e 60 ngā huarahi hei hanga i tētahi tau mati-3 mai i ngā mati 1, 2, 3, 4, me te 5, me te kore e tāruatia tētahi mati.
Tauira Pātai 5
He raruraru:
E ono ngā kaitākaro, arā, a A, a B, a C, a D, a E, me a F. Ka whakaritea rātou kia rite ki ngā mea e toru o runga mō te kēmu. E hia ngā huarahi ka taea te whakarite i ngā kaitākaro tokotoru?
Kōrero:
I konei ka tonoa mātou kia whakarite i ngā kaitākaro e 3 i roto i tētahi raupapa motuhake mai i te katoa o ngā kaitākaro e 6. Ko te tātai i whakamahia ko te whakarōpūtanga \( P(n, r) \) ko \( n = 6 \) me \( r = 3 \):
\[ P(6, 3) = \frac{6!}{(6-3)!} = \frac{6!}{3!} = \frac{6 \times 5 \times 4 \times 3!}{3!} = 6 \times 5 \times 4 = 120 \]
Nō reira, e 120 ngā huarahi hei whakarite i te 3 o ngā kaitākaro 6 ki tētahi raupapa motuhake.
Tauira Pātai 6
He raruraru:
Tātaihia e hia ngā whakarerekētanga o te kupu “WHARE WĀNANGA” kia noho tata tonu ai ngā oropuare tetahi ki tetahi.
Kōrero:
Tekau mā tahi ngā reta o te kupu "WHARE WĀNANGA", ā, ko ngā oropuare ko U, I, E, I, A. Whakaarohia tēnei rōpū oropuare hei wāhanga kotahi.
Nō reira, kei a tātou: (UIEIA), N, V, R, S, T, me S (e kiia ana he kotahi te waeine). Kātahi ka whakarite tātou i ēnei waeine e whitu:
\[ P(7) = 7! = 5040 \]
Heoi, i roto i te rōpū reo (UIEIA), ka taea te whakarite i ēnei ki:
\[ P(5) = 5! = 120 \]
Nō reira, ko ngā huringa katoa ko:
\[ 7! \whakareatia te 5! = 5040 \whakareatia te 120 = 604800 \]
Nō reira, e 604800 ngā huarahi hei hanga i te kupu "WHARE WĀNANGA" ina noho tata tonu ngā oropuare katoa tetahi ki tetahi.
Whakamutunga
Ko te whakarōpūtanga te whakaritenga o ngā mea, o ngā huinga rānei i roto i tētahi raupapa motuhake, ā, he maha ngā whakamahinga o tēnei ariā i roto i ngā momo mara, tae atu ki te pāngarau, te pūtaiao rorohiko, me ngā tatauranga. Mā te tautuhi me te whakatinana i te tātai tika, ka taea e tātou te tatau ngāwari i te maha o ngā whakaritenga ka taea.
Mā ngā tauira kua whakaratohia e whakaatu te mahi a ngā tātai whakatautau me te whakamahinga i roto i ngā āhuatanga maha. He mea nui te māramatanga hōhonu ki ngā whakatautau hei whakaoti rapanga matatini, ā, he mea tino nui hoki hei whakawhanake i te arorau whakaoti rapanga.
