kombi nato rika
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Matematika III. 1.KombinatorikaProf. Dr. Závoti , JózsefTRANSCRIPT
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Created by XMLmind XSL-FO Converter.
Matematika III. 1.
Kombinatorika
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Created by XMLmind XSL-FO Converter.
Matematika III. 1. : Kombinatorika
- 4.1.2-08/1/A-2009- -
A
v 1.0
Publication date 2010 -
Kivonat
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iii Created by XMLmind XSL-FO Converter.
Tartalom
1. Kombinatorika ................................................................................................................................ 1 ........................................................................................................................ 1 ...................................................................................................................... 1
............................................................................. 1 ...................................................................................... 3
........................................................................................................................... 4 .................................................................................. 4
........................................................................................... 5 4. 1. ..................................................................................................................... 6
............................................................................ 6 ..................................................................................... 8
........................................................................................ 9 ................................................................................................ 9
5.2. 1.5.2 Pascal- .......................................................................................... 11 ........................................................................................ 12
................................................................................................................ 13
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1 Created by XMLmind XSL-FO Converter.
1. fejezet - Kombinatorika
Jelen modul a M
azok
a.
b.
Adott
a, b, c
Lexikografikus sorrend:
1. Megkeresni az
2.
3. elemek !
4. A megmaradt
1234 2134 3124 4123
1243 2143 3142 4132
1324 2314 3214 4213
1342 2341 3241 4231
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Kombinatorika
2 Created by XMLmind XSL-FO Converter.
1423 2413 3412 4312
1432 2431 3421 4321
, ahol .
Hasznos a Stirling-
n-1
10! - 9! = 9! (10-
9 9 8 7 6 5 4 3 2 1
-
Kombinatorika
3 Created by XMLmind XSL-FO Converter.
x y x y x y x y x
2.2. 1.2.2
1 2 s
Jele:
uk az a, b, b
n
azonos elemhez
azonos elemhez
...
azonos elemhez
-
Kombinatorika
4 Created by XMLmind XSL-FO Converter.
3:
-
-
kapjuk.
Jele:
:
a b c d
ab ac ad bc bd cd
ba ca da cb db dc
-
1 2 2 1 3 1 n 1
1 3 2 3 3 2 n 2
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Kombinatorika
5 Created by XMLmind XSL-FO Converter.
1 4 2 4 3 4
...
- kapjuk.
Jele:
a, b, c, d elemek
a b c d
aa ba ca da
ab bb cb db
ac bc cc dc
ad bd cd dd
-
n
-
Kombinatorika
6 Created by XMLmind XSL-FO Converter.
n
2 1 2 2 2 n
...
n 1 n 2 n n
A k-
1, 2 elemek 4-
1111 1112 1122 1222 2222
1121 1212 2122
1211 1221 2212
2111 2112 2221
2121
2211
-
-
Kombinatorika
7 Created by XMLmind XSL-FO Converter.
-
Jele:
a, b, c, d elemek 2-
ab bc cd
ac db
ad
-
Az n elem k- -k ki
Az
Legyen
Ha , akkor .
-
Kombinatorika
8 Created by XMLmind XSL-FO Converter.
1, 2, 3, 4, 5 elemek 3-
+ + + - -
+ + - + -
+ + - - +
+ - + + -
+ - + - +
+ - - + +
- + + + -
- + + - +
- + - + +
- - + + +
1 2 3
1 2 4
1 2 5
1 3 4
1 3 5
1 4 5
2 3 4
2 3 5
2 4 5
3 4 5
-
Jele:
a, b, c, d elemek 2-
aa
ab
ac
ad
bb
bc
bd
cc
cd
dd
-
-
Kombinatorika
9 Created by XMLmind XSL-FO Converter.
Az n elem k- - -
-
1 2 3 4 5 6
+ + + + - -
+ + + - + -
+ + + - - +
+ + - + + -
+ + - + - +
+ + - - + +
+ - + + + -
+ - + + - +
+ - + - + +
+ - - + + +
- + + + + -
- + + + - +
- + + - + +
- + - + + +
- - + + + +
1 1 1 1
1 1 1 2
1 1 1 3
1 1 2 2
1 1 2 3
1 1 3 3
1 2 2 2
1 2 2 3
1 2 3 3
1 3 3 3
2 2 2 2
2 2 2 3
2 2 3 3
2 3 3 3
3 3 3 3
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Kombinatorika
10 Created by XMLmind XSL-FO Converter.
-ahol .
a. A polinom
b. Minden tagban az
i. A bk
a
an
Ha a- b-
Ha a- b-
.
-
Kombinatorika
11 Created by XMLmind XSL-FO Converter.
5.2. 1.5.2 Pascal-
A Pascal-
A Pascal-
1. -
2. -
3. -
4.
1.
2. =
3. -et!
4. -et!
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Kombinatorika
12 Created by XMLmind XSL-FO Converter.
m n k
k-1 darab
Legyen adott egy
Ekkor az
Legyen
Az