Ihe Nlereanya nke Ajụjụ Mkparịta ụka Nchikota
Ngwakọta bụ echiche dị mkpa na ozizi na ọnụọgụgụ ihe gbasara ohere, nke na-enye anyị ohere ịgụta ọnụọgụgụ ụzọ isi họrọ ihe site na otu setịpụ n'enweghị ihe gbasara usoro. Na njikọta, usoro nhọrọ anaghị agbanwe agbanwe, n'adịghị ka permutations, ebe usoro nhọrọ dị mkpa. A na-eji njikọta eme ihe n'ọtụtụ ubi dị iche iche, site na sayensị kọmputa na bayoloji ruo na akụnụba na mgbakọ na mwepụ. Isiokwu a ga-akọwa echiche bụ isi nke njikọta ma nye ọtụtụ ihe atụ nsogbu na mkparịta ụka iji nyere gị aka ịghọta.
Echiche Isi nke Nchikota
Tupu anyị agaa n'ihu na nsogbu ihe atụ, ka anyị malite site n'ịghọta ihe dị mkpa gbasara njikọta.
A na-egosi njikọta nke ihe \(n\) e were \(r\) n'otu oge site na ihe e dere \(\binom{n}{r}\) ma ọ bụ \(\mathbf{C}(n, r)\). Usoro maka ịgbakọ njikọta ahụ bụ:
\[ \binom{n}{r} = \frac{n!}{r!(nr)!} \]
– \(n \) bụ mkpokọta ọnụọgụgụ ihe dị.
– \( r \) bụ ọnụọgụ ihe ndị a ga-ahọrọ.
– ! (factorial) bụ ọrụ mgbakọ na mwepụ nke na-amụba ọnụọgụgụ site na ọnụọgụgụ niile dị n'okpuru ya ruo 1.
Ajụjụ na Mkparịta ụka Ihe Nlereanya
Ajụjụ Ihe Nlereanya nke 1: Ịhọrọ otu otu n'ime otu ụmụ akwụkwọ
Ajụjụ:
Ụmụ akwụkwọ iri nọ na klas. Olee ụzọ ole ka a ga-esi họrọ ụmụ akwụkwọ anọ si na klas ahụ isonye na asọmpi ahụ?
Azịza:
Iji usoro njikọta:
\[ \binom{10}{4} = \frac{10!}{4! \cdot (10-4)!} = \frac{10!}{4! \cdot 6!} \]
Ịgbakọ ọnụọgụgụ ihe:
– \( 10! = 10 \ugboro 9 \ugboro 8 \ugboro 7 \ugboro 6! \)
– \( 4! = 4 \ugboro 3 \ugboro 2 \ugboro 1 = 24 \)
– \(6! \) dịlarị n'elu, yabụ ọ nwere ike ịkagbu 6! na \(10!\).
Ya mere:
\[ \binom{10}{4} = \frac{10 \ugboro 9 \ugboro 8 \ugboro 7}{4 \ugboro 3 \ugboro 2 \ugboro 1} = \frac{5040}{24} = 210 \]
Ya mere, e nwere ụzọ iri isi họrọ ụmụ akwụkwọ atọ site na ụmụ akwụkwọ ise.
Ajụjụ Ihe Nlereanya nke Abụọ: Nchikota Site na Iji Setịpụ Kaadị
Ajụjụ:
Site na kaadị 52, ụzọ ole ka e si ahọrọ aka poker nke nwere kaadị 5?
Azịza:
Iji usoro njikọta:
\[ \binom{52}{5} = \frac{52!}{5! \cdot (52-5)!} = \frac{52!}{5! \cdot 47!} \]
Ịgbakọ ọnụọgụgụ ihe:
– \( 52! = 52 \ugboro 51 \ugboro 50 \ugboro 49 \ugboro 48 \ugboro 47!)
– \( 5! = 5 \ugboro 4 \ugboro 3 \ugboro 2 \ugboro 1 = 120 \)
– \(47! \) dịlarị n'elu, yabụ ọ nwere ike ịkagbu 47! na \(52!\).
Ya mere:
\[ \binom{52}{5} = \frac{52 \ugboro 51 \ugboro 50 \ugboro 49 \ugboro 48}{5 \ugboro 4 \ugboro 3 \ugboro 2 \ugboro 1} = \frac{311875200}{120} = 2598960 \]
Ya mere, e nwere ụzọ 2,598,960 iji họrọ kaadị 5 site na kaadị 52.
Ihe atụ nke 3: Ịgbakọ njikọta maka nhọrọ otu egwuregwu
Ajụjụ:
Otu egwuregwu basketball nwere ndị egwuregwu iri na ise. Ọ bụrụ na onye nkuzi chọrọ ịhọrọ ndị egwuregwu ise dịka ndị mbido, ụzọ ole ka a ga-esi mee nke a?
Azịza:
Iji usoro njikọta:
\[ \binom{15}{5} = \frac{15!}{5! \cdot (15-5)!} = \frac{15!}{5! \cdot 10!} \]
Ịgbakọ ọnụọgụgụ ihe:
– \( 15! = 15 \ugboro 14 \ugboro 13 \ugboro 12 \ugboro 11 \ugboro 10!)
– \( 5! = 5 \ugboro 4 \ugboro 3 \ugboro 2 \ugboro 1 = 120 \)
– \(10! \) dịlarị n'elu, yabụ ọ nwere ike ịkagbu 10! na \(15!\).
Ya mere:
\[ \binom{15}{5} = \frac{15 \ugboro 14 \ugboro 13 \ugboro 12 \ugboro 11}{5 \ugboro 4 \ugboro 3 \ugboro 2 \ugboro 1} = \frac{360360}{120} = 3003 \]
Ya mere, e nwere ụzọ 3,003 isi họrọ ndị egwuregwu ise site na ndị egwuregwu iri na ise.
Ajụjụ Ihe Nlereanya nke 4: Nchikota na Bayọlọji
Ajụjụ:
N'ogige, e nwere ụdị okooko 8 dị iche iche. Olee ụzọ ole ka a ga-esi họrọ ụdị okooko 3 iji hazie okooko?
Azịza:
Iji usoro njikọta:
\[ \binom{8}{3} = \frac{8!}{3! \cdot (8-3)!} = \frac{8!}{3! \cdot 5!} \]
Ịgbakọ ọnụọgụgụ ihe:
– \( 8! = 8 \ugboro 7 \ugboro 6 \ugboro 5! \)
– \( 3! = 3 \ugboro 2 \ugboro 1 = 6 \)
– \(5! \) dịlarị n'elu, yabụ ọ nwere ike ịkagbu 5! na \(8!\).
Ya mere:
\[ \binom{8}{3} = \frac{8 \ugboro 7 \ugboro 6}{3 \ugboro 2 \ugboro 1} = \frac{336}{6} = 56 \]
Ya mere, e nwere ụzọ iri ise na isii isi họrọ ụdị okooko atọ site na ụdị okooko asatọ.
Ajụjụ Ihe Nlereanya nke 5: Njiko na Ịwulite Ndị otu
Ajụjụ:
I nwere akwụkwọ iri na abụọ dị iche iche n'elu shelf akwụkwọ. Olee ụzọ ole ị ga-esi họrọ akwụkwọ ise n'ime akwụkwọ iri na abụọ ahụ?
Azịza:
Iji usoro njikọta:
\[ \binom{12}{5} = \frac{12!}{5! \cdot (12-5)!} = \frac{12!}{5! \cdot 7!} \]
Ịgbakọ ọnụọgụgụ ihe:
– \( 12! = 12 \ugboro 11 \ugboro 10 \ugboro 9 \ugboro 8 \ugboro 7!)
– \( 5! = 5 \ugboro 4 \ugboro 3 \ugboro 2 \ugboro 1 = 120 \)
– \(7! \) dịlarị n'elu, yabụ ọ nwere ike ịkagbu 7! na \(12!\).
Ya mere:
\[ \binom{12}{5} = \frac{12 \ugboro 11 \ugboro 10 \ugboro 9 \ugboro 8}{5 \ugboro 4 \ugboro 3 \ugboro 2 \ugboro 1} = \frac{95040}{120} = 792 \]
Ya mere, e nwere ụzọ 792 isi họrọ akwụkwọ ise n'ime akwụkwọ iri na abụọ.
Mmechi
Ngwakọta bụ echiche bara uru nke ukwuu n'ọtụtụ ngalaba dị iche iche, ọkachasị mgbe anyị chọrọ ịma ọnụọgụ ụzọ isi họrọ ihe n'echeghị usoro. Site na iji usoro njikọta bụ isi \(\binom{n}{r}\), anyị nwere ike ịgbakọ ha ngwa ngwa na mfe, ma ọ bụrụhaala na anyị ghọtara otu factorials si arụ ọrụ. Site na ihe atụ ndị dị n'elu, anyị na-atụ anya inwe nghọta zuru oke na nke doro anya nke njikọta. Gaa n'ihu na-eme ihe na ọtụtụ nsogbu iji mee ka nghọta gị sikwuo ike.
