Lipotso tsa Mohlala oa Puisano ea Combinatorics
Combinatorics ke lekala la lipalo le ithutang ho bala, ho hlophisa le dibopeho tse ka bang teng tsa dihlopha tsa dielemente. Combinatorics e na le ditshebediso tsa bohlokwa mafapheng a fapaneng, ho kenyeletswa saense ya dikhomphutha, dipalopalo, baeloji le moruo. Sehloohong sena, re tla buisana ka mehlala e mmalwa le dipuisano tsa yona tse amanang le combinatorics, e leng se tshepahalang hore se tla fana ka kutlwisiso e betere ya dikgopolo tsa motheo le ditshebediso tsa combinatorics.
Potso ea 1: Phetoho
Potso:
Libuka tse 5 tse fapaneng li ka hlophisoa ka litsela tse kae shelofong?
Puisano:
Phetoho ke tlhophiso ea lintho ka tatellano e hlophisitsoeng. Ha tatellano e le ea bohlokoa, re sebelisa phetoho. Moelelong oa bothata bona, re na le libuka tse hlano tse fapaneng tseo re lokelang ho li hlophisa. Palo ea litsela tsa ho hlophisa libuka tsena tse hlano ke:
\[ 5! = 5 \makgetlo a 4 \makgetlo a 3 \makgetlo a 2 \makgetlo a 1 = 120 \]
Kahoo, ho na le litsela tse 120 tsa ho hlophisa libuka tse 5 tse fapaneng shelofong.
Potso ea 2: Motsoako
Potso:
Ho tloha ho batho ba 10, ho na le litsela tse kae tsa ho theha sehlopha sa batho ba 4?
Puisano:
Motswako ke kgetho ya dintho moo tatellano e seng ya bohlokwa. Foromo ya motswako ke:
\[ \binom{n}{k} = \frac{n!}{k!(nk)!} \]
Moelelong oa bothata bona, \( n = 10 \) le \( k = 4 \). Kahoo,
\[ \binom{10}{4} = \frac{10!}{4! \makgetlo (10-4)!} = \frac{10!}{4! \makgetlo a 6!} \]
Rea tseba hore \( 10! = 10 \makgetlo a 9 \makgetlo a 8 \makgetlo a 7 \makgetlo a 6! \), ebe
\[ \binom{10}{4} = \frac{10 \times 9 \times 8 \times 7 \times 6!}{4! \times 6!} = \frac{10 \times 9 \times 8 \times 7}{4 \times 3 \times 2 \times 1} = 210 \]
Kahoo, ho na le mekhoa e 210 ea ho theha sehlopha sa batho ba 4 ho ba 10.
Potso ea 3: Liphetolelo tse nang le Pheta-pheto
Potso:
Ho na le litsela tse kae tsa ho hlophisa lentsoe "LEVEL"?
Puisano:
Lentsoe "LEVEL" le na le litlhaku tse 5, tseo tse ling tsa tsona li phetoang (L habeli le E habeli). Foromo ea phetoho e nang le pheta-pheto ke:
\[ \frac{n!}{n_1! \times n_2! \times \ldots \times n_k!} \]
Moelelong oa bothata bona, \( n = 5 \), \( n_1 = 2 \) bakeng sa tlhaku L, le \( n_2 = 2 \) bakeng sa tlhaku E. Kahoo,
\[ \frac{5!}{2! \times 2!} = \frac{5 \times 4 \times 3 \times 2 \times 1}{2 \times 1 \times 2 \times 1} = \frac{120}{4} = 30 \]
Kahoo, ho na le litsela tse 30 tsa ho hlophisa lentsoe "LEVEL".
Potso ea 4: Motsoako le Pheta-pheto
Potso:
Ho na le litsela tse kae tsa ho khetha lipompong tse 3 ho tsoa mefuteng e 5 e fapaneng ea lipompong tse lumelloang ho pheta-pheta?
Puisano:
Motsoako le ho pheta-pheta ho sebelisoa foromo e latelang:
\[ \binom{n+r-1}{r} \]
Moelelong oa bothata bona, \( n = 5 \) (mefuta ea lipompong) le \( r = 3 \) (palo ea lipompong tse khethiloeng). Kahoo,
\[ \binom{5+3-1}{3} = \binom{7}{3} = \frac{7!}{3! \makgetlo a 4!} \]
Ho tseba \( 7! = 7 \makgetlo a 6 \makgetlo a 5 \makgetlo a 4! \), ebe
\[ \binom{7}{3} = \frac{7 \times 6 \times 5 \times 4!}{3! \times 4!} = \frac{7 \times 6 \times 5}{3 \times 2 \times 1} = 35 \]
Kahoo, ho na le mekhoa e 35 ea ho khetha lipompong tse 3 ho tsoa mefuteng e 5 e fapaneng ea lipompong tse lumelloang ho pheta-pheta.
Potso ea 5: Molao-motheo oa ho Eketsa
Potso:
Ho na le litsela tse kae tsa ho khetha tholoana e le 'ngoe ka basketeng e nang le liapole tse 3, lilamunu tse 2 le libanana tse 5?
Puisano:
Molao-motheo oa ho eketsa o bolela hore haeba ho na le litsela tse 'maloa tsa ho etsa ketso, joale palo eohle ea litsela ke kakaretso ea litsela tseo kaofela. Moelelong oa bothata bona,
– Ho na le litsela tse 3 tsa ho khetha apole e le 'ngoe.
– Ho na le litsela tse peli tsa ho khetha lamunu e le 'ngoe.
– Ho na le litsela tse 5 tsa ho khetha banana e le 'ngoe.
Litsela tsohle:
\[ 3 + 2 + 5 = 10 \]
Kahoo, ho na le litsela tse 10 tsa ho khetha tholoana e le 'ngoe ka basketeng.
Potso ea 6: Molao-motheo oa ho Atisa
Potso:
Ho na le litsela tse kae tsa ho khetha hempe e le 'ngoe ho tsoa likhethong tse 4 le para e le 'ngoe ea borikhoe ho tsoa likhethong tse 3?
Puisano:
Molao-motheo oa katiso o bolela hore haeba ho na le mekhoa e mengata ea ho etsa ketso ea pele le mekhoa e mengata ea ho etsa ketso ea bobeli, joale palo eohle ea litsela tsa ho etsa liketso ka bobeli ke sehlahisoa sa litsela tsa ho etsa ketso ka 'ngoe.
Moelelong oa potso ena,
– Ho na le litsela tse 4 tsa ho khetha hempe e le 'ngoe.
– Ho na le litsela tse 3 tsa ho khetha borikhoe bo le bong.
Litsela tsohle:
\[ 4 \makhetlo a 3 = 12 \]
Kahoo, ho na le litsela tse 12 tsa ho khetha hempe e le 'ngoe le borikhoe bo le bong.
Qetello
Combinatorics, e le lekala la lipalo, e fana ka mekhoa le mehopolo e mengata e mengata bakeng sa ho bala le ho hlophisa lintho tse fapaneng. Ho tloha ho permutations le ho kopanya ho ea ho melao-motheo ea ho eketsa le ho atisa, likhopolo tsena li sebelisoa khafetsa lits'ebetsong tse fapaneng tse sebetsang. Ka ho utloisisa mehlala le lipuisano tse kaholimo, ho tšeptjoa hore babali ba tla khona ho sebelisa likhopolo tsa combinatorics maemong a rarahaneng haholoanyane le ho ntlafatsa tsebo ea bona ea ho rarolla mathata lipalo le mafapheng a mang.
