Ngā Tauira Pātai e Matapaki ana i ngā Ture mō te Whakakī i ngā Wāhi Noho
Ko te ture whakakī-wāhi, te ture whakanoho rānei, he ariā taketake i roto i te pāngarau me te tūponotanga e tino whai hua ana i roto i ngā āhuatanga maha. E whakamahia ana tēnei ture i roto i te horopaki o te whakarite i ngā mea i roto i tētahi raupapa motuhake, i roto rānei i ngā whakaritenga rerekē. I roto i tēnei tuhinga, ka matapakihia e mātou ētahi tauira raruraru e pā ana ki te ture whakakī-wāhi, me te whakarato i ngā otinga taipitopito mō ia mea.
Pendahuluan
Ko te whakakī-ātea he tikanga noa e whakamahia ana i roto i te mahi whakakotahi, he mara pāngarau e ako ana i te whakaritenga, te whakakotahi, me te kōwhiringa o ngā mea. Ko tētahi o ngā mātāpono matua o te mahi whakakotahi ko te ture whakarea, e kī ana mēnā he maha ngā wāhanga i roto i tētahi tukanga, ā, he maha ngā kōwhiringa kei ia wāhanga, ka taea te kimi i te tapeke o ngā whakaritenga ka taea mā te whakarea i te maha o ngā kōwhiringa i ia wāhanga.
Hei tauira, mēnā e rua ā tātou wāhanga, kei reira ngā kōwhiringa \(m\) i te wāhanga tuatahi, ā, kei te wāhanga tuarua ngā kōwhiringa \(n\), ko te tapeke o ngā whakaritenga ka taea ko \(m \times n\).
Me whakamahi tātou i tēnei ariā hei whakaoti rapanga tauira.
Tauira 1: Te Whakarite Pukapuka ki runga i te Whata
Pātai:
E rima ngā pukapuka rerekē me tētahi whata pukapuka me ngā wāhi e rima hei whakakī. E hia ngā huarahi ka taea te whakarite i ngā pukapuka e rima ki runga i te whata?
Kōrero:
I tēnei wā, me whakarite e tātou ngā pukapuka e rima ki roto i ngā wāhi rerekē e rima. He raruraru whakarōpū tēnei nā te mea he mea nui te raupapatanga. Ka taea e tātou te whakamahi i te ture whakakī-wāhi, i te ture whakarea rānei hei whakaoti i tēnei raruraru.
1. Mō te rūma tuatahi, e rima ā mātou whiringa pukapuka.
2. I muri i te whakatakotoranga o tētahi pukapuka ki te rūma tuatahi, e whā ngā kōwhiringa pukapuka e toe ana mō te rūma tuarua.
3. Mō te rūma tuatoru, e toru ā mātou kōwhiringa pukapuka e toe ana, ā, pērā tonu.
Ko te whārite mō te tapeke o ngā tautuhinga ko:
\[ 5 \whakarea 4 \whakarea 3 \whakarea 2 \whakarea 1 = 5! = 120 \]
Nō reira, e 120 ngā huarahi hei whakarite i ngā pukapuka e rima.
Tauira 2: Te Hanga Kupu mai i ngā Reta Rerekē
Pātai:
E hia ngā kupu rerekē ka taea te hanga mā te whakamahi i ngā reta katoa o te kupu "PANGARANGI", me te kore e whakahuatia anō?
Kōrero:
Tuatahi, me kite tātou e hia ngā reta kei roto i te kupu "MATHEMATICS". Tekau mā tahi ngā reta, ko ētahi he mea tāruarua. Ko ngā reta e tāruaruatia ana ko:
– M tae atu ki te 2
– Tae atu ki te 3
– T tae atu ki te 2
– Kotahi te putanga mai o ngā reta kē atu (E, I, K).
Ka whakamahia e mātou te tātai whakarerekētanga mō ngā huānga kua tāruatia, arā:
\[ \frac{n!}{n_1! \times n_2! \times \ldots \times n_k!} \]
ko \( n \) te tapeke o ngā huānga (ngā reta) ā, ko \( n_1, n_2, \ldots, n_k \) te maha o ngā tāruaruatanga o ia huānga motuhake.
Me te kupu "PANGARANGI":
\[ n = 11, n_1 = 2 \text{ (M)}, n_2 = 3 \text{ (A)}, n_3 = 2 \text{ (T)}, n_4 = 1 \text{ (E)}, n_5 = 1 \text{ (I)}, n_6 = 1 \text{ (K)} \]
Nā, ko te maha o ngā kupu ka taea te hanga ko:
\[ \frac{11!}{2! \ngā 3! \ngā 2! \ngā 1! \ngā 1! \ngā 1!} = \frac{39916800}{2 \ngā 6 \ngā 2 \ngā 1 \ngā 1 \ngā 1} = \frac{39916800}{24} = 1663200 \]
E 1,663,200 ngā kupu rerekē ka taea te hanga.
Tauira 3: Te Whakatau i te Tau o ngā Huinga i roto i te Martabak
Pātai:
E rima ngā momo whakakī e tukuna ana e te kaihoko martabak (tīhi, tiakarete, pīnati, panana, me te karepe maroke). Mēnā e hiahia ana te kaihoko ki te whiriwhiri i te toru o ngā whakakī e rima mō tā rātou martabak, e hia ngā momo huinga rerekē ka taea e rātou te whiriwhiri?
Kōrero:
He rapanga whakakotahi tēnei, ehara i te rapanga whakahurihuri, nā te mea kāore he hiranga o te raupapa. Ka whakamahia e mātou te tātai whakakotahi:
\[ C(n, k) = \frac{n!}{k!(nk)!} \]
ko \( n \) te tapeke o ngā kōwhiringa, ā, ko \( k \) te maha o ngā kōwhiringa i tangohia.
Mō tēnei take, \( n = 5 \) me \( k = 3 \), nō reira:
\[ C(5, 3) = \frac{5!}{3!(5-3)!} = \frac{5!}{3! \times 2!} = \frac{120}{6 \times 2} = \frac{120}{12} = 10 \]
E 10 ngā huinga rerekē hei whiriwhiri i ngā ihirangi e 3 mai i ngā kōwhiringa e 5.
Tauira 4: Te Whakaritenga a te Kaiuru i roto i tētahi Whakataetae
Pātai:
E waru ngā tāngata e whai wāhi ana ki tētahi reihi oma. E hia ngā huarahi ka taea ai te whakanoho i ngā tāngata e toru o runga ki te whakaoti?
Kōrero:
He raruraru whakatauira tēnei, kāore he tāruarua, nā te mea he mea nui te raupapatanga o te tūnga. Ka whakamahia e mātou te tātai whakatauira:
\[ P(n, k) = \frac{n!}{(nk)!} \]
Mō tēnei take, \( n = 8 \) me \( k = 3 \), kātahi:
\[ P(8, 3) = \frac{8!}{(8-3)!} = \frac{8!}{5!} = \frac{40320}{120} = 336 \]
Nō reira, e 336 ngā huarahi hei whakanoho i ngā tūranga matua e toru o te 8 o ngā kaiuru.
I roto i tēnei tuhinga, kua matapakihia e mātou ētahi tauira rapanga me ā rātou otinga mā te whakamahi i ngā ture whakakī-ātea i roto i ngā āhuatanga maha: mai i te whakarite pukapuka ki runga i te whata tae atu ki te whakatau ko wai te toa o tētahi whakataetae. Mā te mārama ki ēnei kaupapa matua ka nui ake tō māia ki te whakaoti rapanga whakakotahi me ngā rapanga tūponotanga ka tūtaki pea koe.
