ΠΛΗ20 ΜΑΘΗΜΑ 1.2
-
Upload
dimitris-psounis -
Category
Documents
-
view
29 -
download
0
description
Transcript of ΠΛΗ20 ΜΑΘΗΜΑ 1.2
-
20 1:
1.2:
-
.
.
0.
1. 1.
2.
2. 1.
2 , 20, 1.2:
1.
2.
3.
4.
5.
.
1.
2.
3.
4.
.
1.
2.
3.
-
.
3 , 20, 1.2:
(-)
-
.
( n).
4 , 20, 1.2:
n, 1 n:
12...)2()1(! = nnnn
:
0!=1
12...)2()1(! = nnnn
1,1
1,)!1(
!=
>
=
n
nnnn
-
.
P(n,k) C(n,k)
5 , 20, 1.2:
P(n,k), n! (n-k)! n0 0kn:
)1...()2()1()!(!),( +=
= knnnnkn
nknP
C(n,k) n! k!(n-k)! n0 0kn
To C(n,k) : :
)!(!!),(
knknknC
=
kn
)!(!!
knkn
kn
=
-
.
:
6 , 20, 1.2:
1: :
=
knn
kn
knk
2: :
+
=
111
kn
kn
kn
-
.
() .
: .
7 , 20, 1.2:
.
, , - .
, . ( 1.2) ( 1.3) ( 1.4)
.
-
. 1.
k () n . : , .
8 , 20, 1.2:
, .
4 () ( )
, .
(.. 1-2 2-1 )
-
. 1. :
9 , 20, 1.2:
( 49 6), ( 52 5), .
. .
-
. 1. ,
:
k n
10 , 20, 1.2:
kn
k n
+
kkn 1
:
( .. )
( .. {1,2,..,6} )
-
. 1. 1.
11 , 20, 1.2:
:1. 2. n ( ).3. k ,
( ( )
:
=
kn
knC ),(
, ,
..
-
. 1. 2.
12 , 20, 1.2:
:1. 2. n ( ).3. k
( ( )
:
+=+
kkn
kknC1),1(
, ..
.
,
.
-
. 2. 1.
13 , 20, 1.2:
49 6 . ;:
:
49 816.983.13...649
==
:
1. ( 1, 8, 44, 12, 19, 23 .. 1, 12, 44, 8, 19, 23)
2. 49 63.
.
-
. 2. 2.
14 , 20, 1.2:
(=) . ;:
:
2167!567!7!77126
=
=
===
=
+ 21
2!5!2!5!2)!27(!222 =====
=
:
1. ( 1,2 2,1)
2. n=6 ( 1,2,3,4,5,6) k=2 .
3. (.. 1 2 )
-
. 2. 3.
15 , 20, 1.2:
; (:
,1,2,3,4,5 6 ):
: :
28278
!6!2!678
!6!2!8
)!28(!2!8
28
2127
=
=
=
=
=
=
+
:1. (
2,5 5,2) 2. 7 ( 0,1,2,3,4,5,6)
2 .3.
(.. 1 2 )
-
. 2. 4.
16 , 20, 1.2:
5 .
:
:
52 960.598.2...5
52==
:
1. 2. n=52
k=5 .3.
.
-
. 2. 5.
17 , 20, 1.2:
2 {1,2,3,4,5,6,7,8}:
:
28...8
==
28...2
==
:
1. ( )
2. n=8 ( ) k=2 .
3. .
-
. 2. 5. (1. )
18 , 20, 1.2:
SOS : k=0,1,2,,n :
nnniinn
in ba
n
nba
nba
nba
in
ba
++
+
=
=+
= 01100 ...10)(
kn
:
:
iba
nbababa
iba
++
+
=
=+=
0 ...10)(
32233021120333
03 33
33
23
13
033)( babbaabababababa
iba ii
i+++=
+
+
+
=
=+
=
niinn
in x
n
nx
nx
nx
nx
in
x
++
+
+
=
=+
= ...2101)1(
2100
++
+
+
=
=+
=
n
nnnn
in iinn
in
...
21011)11(
0
n
n
nnnn2...
210=
++
+
+
-
. 2. 5. (2. )
19 , 20, 1.2:
={1,2,3,4} :
------------------------------------> C(4,0)=1 ---------------------> C(4,1)=4
---> C(4,2)=6 },4,3{},4,2{},3,2{},4,1{},3,1{},2,1{},4{},3{},2{},1{
,
{)(
=AP
---> C(4,2)=6 --------> C(4,3)=4
--------------------------------> C(4,4)=1
:
:
}}4,3,2,1{
},4,3,2{},4,3,1{},4,2,1{},3,2,1{},4,3{},4,2{},3,2{},4,1{},3,1{},2,1{
4244
34
24
14
04
=
+
+
+
+
n
n
nnnn2...
210=
++
+
+
-
. 1.
20 , 20, 1.2:
: :) (= ). .) . .. 5 , 4 6 , 3 ) . . .
:
50 . 5 :
1 .
.
1 .
-
. 2.
21 , 20, 1.2:
:
:
.
.
:
5 , 5 5 . 4 . :
n=3(1 , 1 1 ) k=4. :
C(n+k-1,k)=C(3+4-1,4)=C(6,4)=15 .
-
. 2.
22 , 20, 1.2:
:
.
.
:
5 , 5 2 . 4 . :
:
0 : C(2+4-1,4)=C(5,4) 1 . 3 C(2+3-1,3)=C(4,3) 2 . 2 C(2+2-1,2)=C(3,2)
.
:
C(5,4)+C(4,3)+C(3,2)=5+4+3=12
-
. 3.
23 , 20, 1.2:
:
. -
.
,
. .
:
20 3 , 1 6, 2 9 3 5 . ;:
1 C(20,6) 2 C(14,9) 3 C(5,5) : C(20,6)C(14,9)C(5,5)
! , .
.
-
. 4.
24 , 20, 1.2:
:
.
, :
:
5 , 5 5 . 4 2 :
:
2 : C(2+2-1,2)=C(3,2)=3 3 . C(2+1-1,1)=C(2,1)=2 4 . 1 3+2+1=6
() .
-
.
1 :
25 , 20, 1.2:
=
=
=
=
=
!4)!3)!2)!1)!0)
EDCBA
=
=
=
=
!7)!6)!5)!4)
HGFE
-
.
2 :
26 , 20, 1.2:
=
=
=
=
=
)2,4())1,4())0,4())2,5())3,5()
CECDCCPBPA
=
=
=
=
)3,10())4,4())3,4())2,4()
CHCGCFCE
-
.
3 :
27 , 20, 1.2:
=
knn
kn
-
.
1 A={1,2,3,4,5,6,7,8,9,10}
1. A 5
2. A 3 7
28 , 20, 1.2:
510
2. A 3 7
3. A 4 6
4. 210
-
.
2 6 3 . , 4 2
1. 15.
29 , 20, 1.2:
2. 9.
3. 8.
4. 27.
-
.
3 52 (4 13 ). .
1. 1312.
30 , 20, 1.2:
2. 5 4 .
3. 5 4 4.
4. 5 3 2
452
249
352
-
.
4 2 7 :
1. 2 6 ( )
31 , 20, 1.2:
2. 2 {1,2,3,4,5,6,7}
3. x2 (1+x)7
4. 2 .
-
.
1 4 2 , 5 1 7 0.50 . 3 ;
32 , 20, 1.2:
-
.
2 30 , 30 30 . 10 , . 30 , 30 5 .
33 , 20, 1.2:
-
.
3 100 1 40 , 2 30 , 3 20 4 10 . ;
34 , 20, 1.2:
-
.
4 4 , 4 4 . 4 :1. 2 2. 2 3. 2
35 , 20, 1.2:
-
.
5) coach 11 3-5-2(3 , 5 2 ). , 7 , 8 4 .
) 20 4 2,5,6,7
36 , 20, 1.2:
) 20 4 2,5,6,7 .
.
) 12 . 5 1 , 3 2 4 3 . ;
-
.
6 10
) . , , ,
.
) ;)
37 , 20, 1.2:
) ;