Tauira o ngā Pātai Kōrero Whakakotahi
He ariā nui ngā huinga i roto i te ariā tūponotanga me ngā tatauranga, e āhei ai tātou ki te tatau i te maha o ngā huarahi hei tīpako i ngā mea mai i tētahi huinga me te kore e aro ki te raupapa. I roto i ngā huinga, ka noho tonu te raupapa o te kōwhiringa, he rerekē ki ngā whiriwhiringa, ko te raupapa o te kōwhiringa te mea nui. Ka whakamahia ngā huinga i roto i te whānuitanga o ngā mara, mai i te pūtaiao rorohiko me te koiora ki te ōhanga me te pāngarau. Ka whakamāramahia e tēnei tuhinga te ariā taketake o ngā huinga, ā, ka whakaratohia he tauira rapanga me ngā kōrero hei āwhina i tō māramatanga.
Ariā Taketake o te Whakakotahitanga
I mua i te neke atu ki ngā tauira rapanga, me tīmata tātou me te māramatanga taketake mō ngā huinga.
Ko te huinga o ngā mea \(n\) i tangohia \(r\) i te wā kotahi ka tohua e te tohu \(\binom{n}{r}\) me te \(\mathbf{C}(n, r)\). Ko te tātai mō te tatau i te huinga ko:
\[ \binom{n}{r} = \frac{n!}{r!(nr)!} \]
– Ko te \( n \) te tapeke o ngā mea e wātea ana.
– Ko te \( r \) te maha o ngā mea hei tīpako.
– ! (whakarea) he mahi pāngarau e whakarea ana i tētahi tau ki ngā tau pai katoa i raro iho i taua tau tae atu ki te 1.
Ngā Pātai Tauira me te Kōrero
Tauira Pātai 1: Te Kōwhiri i tētahi Tīma mai i tētahi Rōpū Ākonga
Pātai:
Tekau ngā ākonga i roto i tētahi akomanga. E hia ngā huarahi hei whiriwhiri i ngā ākonga tokowhā o te akomanga hei whai wāhi atu ki te whakataetae?
Kōrero:
Mā te whakamahi i te tātai whakakotahi:
\[ \binom{10}{4} = \frac{10!}{4! \cdot (10-4)!} = \frac{10!}{4! \cdot 6!} \]
Te tatau i te tauwehenga:
– \( 10! = 10 \whakareatia 9 \whakareatia 8 \whakareatia 7 \whakareatia 6! \)
– \( 4! = 4 \whakareatia ki te 3 \whakareatia ki te 2 \whakareatia ki te 1 = 24 \)
– Kei runga rawa kē a \( 6! \) , nō reira ka taea e ia te whakakore i te 6! i roto i te \(10!\).
Nā reira:
\[ \binom{10}{4} = \frac{10 \times 9 \times 8 \times 7}{4 \times 3 \times 2 \times 1} = \frac{5040}{24} = 210 \]
Nō reira, e 210 ngā huarahi hei whiriwhiri i ngā ākonga e 4 mai i ngā ākonga e 10.
Tauira Pātai 2: Ngā Huinga e Whakamahi ana i ngā Huinga Kāri
Pātai:
Mai i tētahi kete kāri e 52, e hia ngā huarahi hei whiriwhiri i tētahi ringa pōka e 5 ngā kāri?
Kōrero:
Mā te whakamahi i te tātai whakakotahi:
\[ \binom{52}{5} = \frac{52!}{5! \cdot (52-5)!} = \frac{52!}{5! \cdot 47!} \]
Te tatau i te tauwehenga:
– \( 52! = 52 \ngā 51 \ngā 50 \ngā 49 \ngā 48 \ngā 47! \)
– \( 5! = 5 \whakareatia 4 \whakareatia 3 \whakareatia 2 \whakareatia 1 = 120 \)
– Kei runga rawa kē a \( 47! \) , nō reira ka taea e ia te whakakore i te 47! i roto i te \(52!\).
Nā reira:
\[ \binom{52}{5} = \frac{52 \times 51 \times 50 \times 49 \times 48}{5 \times 4 \times 3 \times 2 \times 1} = \frac{311875200}{120} = 2598960 \]
Nō reira, e 2,598,960 ngā huarahi hei whiriwhiri i ngā kāri e 5 mai i ngā kāri e 52.
Tauira 3: Te Tatau i ngā Huinga mō te Kōwhiringa Tīma Hākinakina
Pātai:
Tekau mā rima ngā kaitākaro o tētahi tīma poitūkohu. Mēnā e hiahia ana te kaiako ki te whiriwhiri i te rima o ngā kaitākaro hei tīmatanga, e hia ngā huarahi ka taea ai tēnei?
Kōrero:
Mā te whakamahi i te tātai whakakotahi:
\[ \binom{15}{5} = \frac{15!}{5! \cdot (15-5)!} = \frac{15!}{5! \cdot 10!} \]
Te tatau i te tauwehenga:
– \( 15! = 15 \ngā 14 \ngā 13 \ngā 12 \ngā 11 \ngā 10! \)
– \( 5! = 5 \whakareatia 4 \whakareatia 3 \whakareatia 2 \whakareatia 1 = 120 \)
– Kei runga rawa kē a \( 10! \) , nō reira ka taea e ia te whakakore i te 10! i roto i te \(15!\).
Nā reira:
\[ \binom{15}{5} = \frac{15 \times 14 \times 13 \times 12 \times 11}{5 \times 4 \times 3 \times 2 \times 1} = \frac{360360}{120} = 3003 \]
Nō reira, e 3,003 ngā huarahi hei whiriwhiri i te 5 kaitākaro mai i te 15 kaitākaro.
Tauira Pātai 4: Ngā Huinga i roto i te Koiora
Pātai:
I roto i tētahi māra, e waru ngā momo putiputi. E hia ngā huarahi ka taea te whiriwhiri i ngā momo putiputi e toru hei hanga i tētahi whakaritenga putiputi?
Kōrero:
Mā te whakamahi i te tātai whakakotahi:
\[ \binom{8}{3} = \frac{8!}{3! \cdot (8-3)!} = \frac{8!}{3! \cdot 5!} \]
Te tatau i te tauwehenga:
– \( 8! = 8 \whakareatia 7 \whakareatia 6 \whakareatia 5! \)
– \( 3! = 3 \whakareatia ki te 2 \whakareatia ki te 1 = 6 \)
– Kei runga rawa kē a \( 5! \) , nō reira ka taea e ia te whakakore i te 5! i roto i te \(8!\).
Nā reira:
\[ \binom{8}{3} = \frac{8 \times 7 \times 6}{3 \times 2 \times 1} = \frac{336}{6} = 56 \]
Nō reira, e 56 ngā huarahi hei whiriwhiri i ngā momo putiputi e 3 mai i ngā momo putiputi e 8.
Tauira Pātai 5: Te Whakakotahitanga i roto i te Hanga Rōpū
Pātai:
Tekau mā rua āu pukapuka rerekē kei runga i te whata pukapuka. E hia ngā huarahi ka taea e koe te whiriwhiri i te rima pukapuka mai i ngā tekau mā rua?
Kōrero:
Mā te whakamahi i te tātai whakakotahi:
\[ \binom{12}{5} = \frac{12!}{5! \cdot (12-5)!} = \frac{12!}{5! \cdot 7!} \]
Te tatau i te tauwehenga:
– \( 12! = 12 \ngā 11 \ngā 10 \ngā 9 \ngā 8 \ngā 7! \)
– \( 5! = 5 \whakareatia 4 \whakareatia 3 \whakareatia 2 \whakareatia 1 = 120 \)
– Kei runga rawa kē a \( 7! \) , nō reira ka taea e ia te whakakore i te 7! i roto i te \(12!\).
Nā reira:
\[ \binom{12}{5} = \frac{12 \times 11 \times 10 \times 9 \times 8}{5 \times 4 \times 3 \times 2 \times 1} = \frac{95040}{120} = 792 \]
Nō reira, e 792 ngā huarahi hei whiriwhiri i ngā pukapuka e 5 mai i roto i ngā pukapuka e 12.
Whakamutunga
He ariā tino whai hua ngā huinga i roto i ngā momo mara, inā koa ka hiahia tātou ki te mōhio ki te maha o ngā huarahi hei whiriwhiri i ngā mea me te kore e aro ki te raupapa. Mā te whakamahi i te tātai huinga taketake \(\binom{n}{r}\), ka taea e tātou te tatau tere me te ngāwari, mena ka mārama tātou ki te mahi a ngā tauwehenga. Mā roto i ngā tauira i runga ake nei, ko te tumanako ka nui ake te māramatanga me te māramatanga mō ngā huinga. Me haere tonu te whakaharatau me ngā momo raruraru hei whakapakari ake i tō māramatanga.
