Mehlala ea Lipotso Tse Buisanang ka Melao ea ho Tlatsa Libaka
Molao oa ho tlatsa sebaka, kapa molao oa ho beha, ke mohopolo oa motheo lipalo le monyetla o thusang haholo maemong a mangata. Molao ona o sebelisoa hangata moelelong oa ho hlophisa lintho ka tatellano e itseng kapa ka litlhophiso tse fapaneng. Sehloohong sena, re tla tšohla mehlala e 'maloa ea mathata a amanang le molao oa ho tlatsa sebaka, re fane ka litharollo tse qaqileng bakeng sa e 'ngoe le e 'ngoe.
Pendahuluan
Ho tlatsa sebaka ke mokhoa o tloaelehileng o sebelisoang ho combinatorics, e leng lefapha la lipalo le ithutang tlhophiso, motsoako le khetho ea lintho. E 'ngoe ea melao-motheo ea combinatorics ke molao oa ho atisa, o bolelang hore haeba ho na le mekhahlelo e 'maloa ts'ebetsong 'me mohato ka mong o na le palo e itseng ea likhetho, joale palo eohle ea litlhophiso tse ka khonehang e ka fumanoa ka ho atisa palo ea likhetho mohatong ka mong.
Mohlala, haeba re na le mekhahlelo e 'meli moo mohato oa pele o nang le likhetho tsa \(m\) le mohato oa bobeli o nang le likhetho tsa \(n\), joale palo eohle ea litlhophiso tse ka khonehang ke \(m \times n\).
A re sebeliseng mohopolo ona ho rarolla mathata a mang a mohlala.
Mohlala oa 1: Ho Hlophisa Libuka Shelofong
Potso:
Ho na le libuka tse 5 tse fapaneng le shelofo ea libuka e nang le libaka tse 5 tsa ho tlatsa. Libuka tse hlano li ka hlophisoa ka litsela tse kae shelofong?
Puisano:
Tabeng ena, re hloka ho hlophisa libuka tse hlano libakeng tse hlano tse fapaneng. Ena ke bothata ba permutation hobane tatellano ea lintho e bohlokoa. Re ka sebelisa molao oa ho tlatsa sebaka kapa molao oa ho atisa ho rarolla bothata bona.
1. Bakeng sa kamore ea pele, re na le likhetho tse 5 tsa libuka.
2. Kamora hore buka e le 'ngoe e beoe ka phapusing ea pele, re setse le likhetho tse 4 tsa libuka bakeng sa phapusing ea bobeli.
3. Bakeng sa kamore ea boraro, re na le likhetho tse 3 tse setseng tsa libuka, joalo-joalo.
Tekanyo ea palo eohle ea litlhophiso ke:
\[ 5 \makgetlo a 4 \makgetlo a 3 \makgetlo a 2 \makgetlo a 1 = 5! = 120 \]
Kahoo, ho na le litsela tse 120 tsa ho hlophisa libuka tse hlano.
Mohlala oa 2: Ho Etsa Mantsoe ka Litlhaku Tse Fapaneng
Potso:
Ke mantsoe a makae a fapaneng a ka bopjwang ka ho sebedisa ditlhaku tsohle tse lentsweng “MALO”, ntle le ho a pheta?
Puisano:
Re lokela ho qala ka ho bona hore na ho na le litlhaku tse kae lentsoeng "LIPALO." Ho na le litlhaku tse 11, tseo tse ling tsa tsona li phetoang. Litlhaku tse phetoang ke tsena:
– M ho fihlela ho 2
– Ho fihlela ho tse 3
– T tse ngata ho fihlela ho tse 2
– Litlhaku tse ling (E, I, K) ka 'ngoe li hlaha hanngoe.
Re sebelisa foromo ea permutation bakeng sa likarolo tse phetoang, e leng:
\[ \frac{n!}{n_1! \times n_2! \times \ldots \times n_k!} \]
moo \( n \) e leng palo yohle ya dielemente (ditlhaku) le \( n_1, n_2, \ldots, n_k \) e leng palo ya diphetoho tsa elemente ka nngwe e ikgethang.
Ka lentsoe "LIPALO":
\[ n = 11, n_1 = 2 \mongolo{ (M)}, n_2 = 3 \mongolo{ (A)}, n_3 = 2 \mongolo{ (T)}, n_4 = 1 \mongolo{ (E)}, n_5 = 1 \mongolo{ (I)}, n_6 = 1 \mongolo{ (K)} \]
Kahoo palo ea mantsoe a ka thehoang ke:
\[ \frac{11!}{2! \makhetlo a 3! \makhetlo a 2! \makhetlo a 1! \makhetlo a 1! \makhetlo a 1!} = \frac{39916800}{2 \makhetlo a 6 \makhetlo a 2 \makhetlo a 1 \makhetlo a 1 \makhetlo a 1} = \frac{39916800}{24} = 1663200 \]
Ho na le mantsoe a fapaneng a 1,663,200 a ka thehoang.
Mohlala oa 3: Ho Fumana Palo ea Metswako ho Martabak
Potso:
Morekisi oa martabak o fana ka likhetho tse hlano tsa ho tlatsa (chisi, tsokolate, linate tsa peanut, banana le morara o omisitsoeng). Haeba moreki a batla ho khetha tse tharo ho tse hlano tsa ho tlatsa martabak ea hae, a ka khetha metsoako e mekae e fapaneng?
Puisano:
Ena ke bothata ba motsoako, eseng phetoho, hobane tatellano ha e bohlokoa. Re sebelisa foromo ea motsoako:
\[ C(n, k) = \frac{n!}{k!(nk)!} \]
moo \( n \) e leng palo yohle ya dikgetho, mme \( k \) e le palo ya dikgetho tse entsweng.
Tabeng ena, \( n = 5 \) le \( k = 3 \), kahoo:
\[ C(5, 3) = \frac{5!}{3!(5-3)!} = \frac{5!}{3! \times 2!} = \frac{120}{6 \times 2} = \frac{120}{12} = 10 \]
Ho na le metsoako e 10 e fapaneng ea ho khetha litaba tse 3 ho tsoa likhethong tse 5.
Mohlala oa 4: Tokisetso ea Bankakarolo Papaling
Potso:
Ho na le barupeluoa ba 8 peisong ea ho matha. Ba 3 ba holimo ba ka beoa ka litsela tse kae?
Puisano:
Ena ke bothata ba phetoho ntle le ho pheta-pheta hobane boemo bo bolela hore tatellano ea lintho e bohlokoa. Re sebelisa foromo ea phetoho:
\[ P(n, k) = \frac{n!}{(nk)!} \]
Tabeng ena, \( n = 8 \) le \( k = 3 \), ebe:
\[ P(8, 3) = \frac{8!}{(8-3)!} = \frac{8!}{5!} = \frac{40320}{120} = 336 \]
Kahoo, ho na le litsela tse 336 tsa ho beha maemo a mararo a holimo a barupeluoa ba 8.
Sehloohong sena, re buisane ka mathata a 'maloa a mehlala le litharollo tsa 'ona re sebelisa melao ea ho tlatsa sebaka maemong a fapaneng: ho tloha ho hlophiseng libuka shelefong ho isa ho fumaneng mohlōli oa tlholisano. Ho utloisisa metheo ena ho tla u fa kholiseho e eketsehileng ho rarolleng mathata a fapaneng a kopanyo le a monyetla ao u ka 'nang ua kopana le 'ona.
