inhlanganisela
I-Pendahuluan
Kumathematika, inhlanganisela ingumqondo oyisisekelo osetshenziswa ukunquma inani lezindlela zokukhetha isethi yezinto ngaphandle kokubheka ukuhleleka kwazo. Inhlanganisela ivame ukusetshenziswa emikhakheni ehlukahlukene yesayensi, kufaka phakathi izibalo, amathuba, kanye nokubala, kanye nokuphila kwansuku zonke, njengokuhlela, imidlalo yamakhadi, nokuningi. Lesi sihloko sizohlola umqondo wokuhlanganiswa ngokujulile, amafomula asetshenziswayo, kanye nezinye zezindlela zokusebenza kwazo ezingokoqobo.
Incazelo Yokuhlanganiswa
Inhlanganisela iyindlela yokukhetha inani lezinto ezivela kusethi ngaphandle kokubheka ukuthi zikhethwe kanjani. Uma kukhona izinto ezahlukene ezingu-\( n \) futhi sifuna ukukhetha izinto ezingu-\( r \) ezivela kusethi, khona-ke inhlanganisela ingavezwa ngenothi \( C(n, r) \) noma \( \binom{n}{r} \).
Ifomula Ehlanganisiwe
Ifomula yokubala inani lezinhlanganisela \( \binom{n}{r} \) ithi:
\[ \binom{n}{r} = \frac{n!}{r! (nr)!} \]
Lapho i-\( n! \) (funda “n factorial”) ingumkhiqizo wazo zonke izinombolo eziqondile kuze kufike ku-\( n \). Isibonelo, i-\( 5! = 5 \izikhathi 4 \izikhathi 3 \izikhathi 2 \izikhathi 1 = 120 \).
Isibonelo Sokubala Okuhlanganisiwe
Ake sithi sinezinto ezi-5 ezahlukene, futhi sifuna ukukhetha izinto ezi-3 kusethi. Zingaki izindlela esingakwenza ngazo lokhu?
Ukusebenzisa ifomula yokuhlanganisa:
\[ \binom{5}{3} = \frac{5!}{3! (5-3)!} = \frac{5!}{3! \cdot 2!} = \frac{120}{6 \cdot 2} = 10 \]
Ngakho-ke, kunezindlela eziyi-10 zokukhetha izinto ezi-3 ezintweni ezi-5 ngaphandle kokunaka i-oda.
Ukuhlanganiswa Kwamathuba
Izinhlanganisela zivame ukusetshenziswa ekutholakaleni kwamathuba ukuze kunqunywe amathuba omcimbi. Isibonelo, emidlalweni yamakhadi, singasebenzisa izinhlanganisela ukuze sibale amathuba okuthola isandla esithile.
Isibonelo 1: Umdlalo Wamakhadi
Ake sithi sifuna ukubala amathuba okuthola ama-Ace amabili kumakhadi amahlanu asetshenziswa edekini ejwayelekile yamakhadi angu-52. Singasebenzisa izinhlanganisela ukubala inani eliphelele lezindlela zokukhetha amakhadi amahlanu kumakhadi angu-52 kanye nenani lezindlela zokukhetha ama-Ace amabili kumakhadi amane.
Izindlela eziphelele zokukhetha amakhadi angu-5 kwangu-52:
\[ \binom{52}{5} = \frac{52!}{5! \cdot 47!} \]
Indlela yokukhetha ama-Ace ama-2 kuma-Ace ama-4:
\[ \binom{4}{2} = \frac{4!}{2! \cdot 2!} = 6 \]
Indlela yokukhetha amakhadi ama-3 kumakhadi angu-48 asele (njengoba amakhadi ama-2 asevele ekhethiwe kuma-Aces ama-4):
\[ \binom{48}{3} = \frac{48!}{3! \cdot 45!} \]
Ngakho-ke, amathuba okuthola ama-Ace amabili esandleni samakhadi amahlanu yilawa:
\[ P = \frac{\binom{4}{2} \cdot \binom{48}{3}}{\binom{52}{5}} \]
Inhlanganisela kuzibalo
Kuzibalo, ukuhlanganiswa kusetshenziswa ekuhlaziyeni okuhlukahlukene, okuhlanganisa ukusampula kanye nokuhlolwa kwe-hypothesis. Ukuhlanganiswa kusiza ekunqumeni inani lezindlela zokukhetha isampula ethile kubantu.
Isibonelo 2: Ukuthatha amasampula
Ake sithi sinabantu abangu-20 futhi sifuna ukukhetha isampula yabantu abangu-4. Singasebenzisa inhlanganisela ukuthola inani lezindlela zokukhetha leso sampula.
\[ \binom{20}{4} = \frac{20!}{4! \cdot 16!} \]
Le nombolo inikeza ulwazi olubalulekile ekuhlaziyweni kwezibalo okuhlukahlukene, njengasekubalweni kwamathuba noma ekulinganisweni kwamapharamitha.
Ithiyori Yokuhlanganiswa Nezinombolo
Ukuhlanganiswa nakho kudlala indima ebalulekile kumbono wezinombolo, ikakhulukazi kumongo wama-binomial coefficients kanye nezinye izici zezibalo. Isibonelo, i-Binomial Theorem isebenzisa ukuhlanganiswa ukuthuthukisa izinkulumo ze-algebraic.
I-Binomial Theorem
I-Binomial Theorem ithi:
\[ (x + y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{nk} y^k \]
Lapha, i-binomial coefficient \( \binom{n}{k} \) ikhombisa inani lezindlela zokukhetha izinto \( k \) ezintweni \( n \).
Izicelo Zokuhlanganiswa Empilweni Yansuku Zonke
Umqondo wokuhlanganiswa awugcini nje ngokubala izibalo, kodwa futhi unezindlela eziningi ezisebenzayo ekuphileni kwansuku zonke.
Isibonelo 3: Izilungiselelo Zeshejuli
Ake sithi sinemisebenzi emi-5 okufanele siyiqede ngesonto, futhi sifuna ukukhetha emi-3 esizoyisebenza ngoMsombuluko. Singasebenzisa inhlanganisela ukuthola inani lezindlela zokukhetha imisebenzi emi-3 kuleyo emi-5.
\[ \binom{5}{3} = 10 \]
Ngakho-ke, kunezindlela eziyi-10 zokuhlela ishejuli.
Isibonelo 4: Ukukhetha Ukudla
Ake sithi sinemenyu enezinketho zokudla eziyi-10, futhi sifuna ukukhetha izitsha ezine zephathi. Singasebenzisa inhlanganisela ukuthola inani lezindlela zokukhetha izitsha ezi-4 kweziyi-10.
\[ \binom{10}{4} = 210 \]
Ngakho-ke, kunezindlela ezingu-210 zokukhetha ukudla okungu-4 kwezingu-10.
Isiphetho
Ukuhlanganiswa kuwumqondo obalulekile kwizibalo futhi kunezinhlelo zokusebenza ezibanzi emikhakheni eyahlukene yesayensi kanye nokuphila kwansuku zonke. Ngokuqonda umqondo wokuhlanganiswa kanye nendlela yokukubala, singaxazulula kalula izinkinga ezahlukahlukene ezihilela ukukhetha izinto ngaphandle kokubheka ukuhleleka. Ukuhlanganiswa kuyasiza ekubalweni kwamathuba, ekuhlaziyweni kwezibalo, embonweni wezinombolo, ngisho nasekuhleleni imisebenzi yansuku zonke. Ngethemba ukuthi lesi sihloko sinikeze ukuqonda okungcono kokuhlanganiswa kanye nokusebenza kwazo ezimweni ezahlukene.
