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ΠΛΗ20 ΕΝΟΤΗΤΑ  1: ΣΥΝ∆ΥΑΣΤΙΚΗ Μάθηµα 1.2: Συ νδυασ οί ∆ηµήτρης Ψούνης

# ΠΛΗ20 ΜΑΘΗΜΑ 1.2

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ΠΛΗ20 ΜΑΘΗΜΑ 1.2

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• 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:

) ;