formuly kombinatoriki

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1)
1!
2!
3!
4!
5!
6!
7!
...
(n
n!
(n
...
(
1
1
1
1
1
1
1
2
2
2
2
2
2
)
n
2
3
3
3
3
3
6
4 24
4 5 120
4 5 6 720
4 5 6 7 5040
1)! 1 2 3 4 5 ... (n 2)( n 1)
1 2 3 4 5 ... ( n 2)( n 1)n
1)! 1 2 3 4 5 ... (n 2)(n 1)n(n 1)
0! 1
2)
n
: Pn
n!
n
:
m
n
n!
(n m)! m!
m
n ?».
m
n
n
0 m n
: Anm
-
(n m 1) ... (n 1)n
»
m
n
Pm
Anm
m
n
Pm
n!
n!
m!
( n m)! m !
( n m)!
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1 2 3 ... (n m)(n m 1) ... (n 1)n
1 2 3 ... (n m)
( n m 1) ... ( n 1)n
Anm
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3)
(a b )
n:
( a b)
0
1
( a b)1
C10a C11b
( a b) 2
C20 a 2
C21a1b1 C22b 2
( a b )3
C30a 3
C31a 2b1 C32 a1b 2
C33b3
( a b) 4
C40 a 4
C41a 3b1 C42 a 2b 2
C43a1b3
C44b 4
( a b )5
C50 a5
C51a 4b1 C52 a 3b 2
C53a 2b3
C54 a1b 4
Cn0a n
Cn1a n 1b1 Cn2a n 2b 2
a b
a2
2ab b 2
a3
3a 2b 3ab 2
a4
b3
4a 3b 6a 2b 2
C55b5
a5
4ab 3
b4
5a 4b 10a 3b 2 10a 2b 3
5ab 4
b5
...
( a b) n
Cn3a n 3b3 ... Cnn 2a 2b n
2
Cnn 1a1b n
1
Cnnb n
...
Cnm
(
:
6
(a b) 6
C60a 6
a6
C61a5b1 C62 a 4b2
6a5b 15a4b 2
C63a3b3 C64 a 2b4
20a 3b 3
15a2b 4
C65a1b5
6ab 5
C66b6
b6
m
n
C62
C103
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: C62 15 ;
2 + 1 = 3-
C95
5 + 1 = 610
3 + 1 = 4-
n!
):
(n m)! m !
: C95
(
(
126 ;
: C103 120 .
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4)
m
A
B
A
n
B
m n
A
m
B
mn
( A; B)
n
.
5)
)
: Pn (
n1
n2
n3 ... nk
n
-
n
3-
n3
n!
,
n1! n2! n3! ... nk !
)
-
n1
, k-
n2
nk
ni
.
:
m
n(
:«
m
n m 1
)
(n m 1)!
(n 1)! m!
n
m
,
m n
m
.
: Anm(
)
nm
n
:
m
.
n
m
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-
.