ΣΥΝΔΥΑΣΤΙΚΗ

: thanasiskopadis.blogspot.com

[1]


[2]

,

: ()

()()=

, , () .

,

.

: , ,

.

1) () :

1

4 : ,,,

3 : ,,

) ; ) 2 ;

) : 4+3=7 ( , , , , , ) ) : 43=12 ( , , , , , , , , , , )


[3]

,

.

, .

- 1 1 , 2 2 ,.., , 12..

-.

2

0,1,2,3,4,5 6, ;

6 , 1,2,3,4,5 6. 7 , 7 0,1,2,3,4,5 6. 67=42 .

1) .

,

;

2)

n!=1 2 3 n

3!= 6321 = , 4!=24, 1!=1. 0!=1

, .

7654321

7654321!5!7

=

=

n

k

, :


[4]

n n!k k!(n - k)!

=

!4!3

!737

=

!4!3!7

37

=

!3765

= 35321765=

=

!4!3

!737

=

47

.

n

k

n

n - k

.

, k 0,1, n n-1,

n= 1

0

,

n= 1

n

,

n= n

1

,

n= n

n -1

( )

nk

: nkn!

(n k)!= , n k

3

) (n 1)(n 1)!(n 1)!

+

) : 2n! (n 1)! (n 2)! n!(n 2)+ + + + = +

) (n 1)! 1 2 3 (n 1) n (n 1) (n 1)!n(n 1)+ = + = +

: (n 1)(n 1)! (n 1)(n 1)! (n 1)

(n 1)! (n 1)!n(n 1) n(n 1)

= =+ + +


[5]

) n! (n 1)! (n 2)! n! n!(n 1) n!(n 1)(n 2)+ + + + = + + + + + =

[ ]n! 1 (n 1) (n 1)(n 2)= + + + + + =

[ ]n! (n 2) (n 1)(n 2)= + + + + =

2n!(n 2)(1 n 1) n!(n 2)= + + + = +

4

: n 1 n 1 n k k 1 k

+ = , n k+1 , n

:

n 1 n 1 (n 1)! (n 1)! k k 1 k!(n 1 k)! (k 1)!(n k)!

+ = + =

(n 1)! (n 1)!(k 1)!k(n 1 k)! (k 1)!(n k 1)!(n k)

= + =

(n 1)! 1 1(k 1)!(n k 1)! k n k

= + =

n(n 1)! n n!k(k 1)!(n k 1)! k(n k) k!(n k)!

= = =

5

: n 2n2 22 50 + =

n 2 2n 2 n 1 . n 2

:

n 2n2 2

n! (2n)!2 50 2 50(n 2)! (2n 2)!+ = + =

(n 2)!(n 1)n (2n 2)!(2n 1)2n2 50 2(n 1)n 50 (2n 1)2n(n 2)! (2n 2)!

+ = + =

2 2 2 22n 2n 50 4n 2n 2n 50 n 25 n 5 + = = = = , n 2


[6]

2) :

) n!(n 1)! ) (n 1)!

(n 1)!(n 1)+

+ ) (n 3)!(n 1)!

n!(n 2)! +

3) n , :

) n! 6(n 3)!= ) n n5 3

=

) n n4 3 2=

4) n n10 215 3

=

,

n

6

, n

5) n , m n m 1 + , : n n 1

m nm m 1

=

6) n 10m

=

nm 60= , n m.

3) n 1 2 na ,a ,...., a ,

.

, a,b,c , :

(a,b,c) , (a,c,b) , (b,a,c) , (b,c,a) , (c,a,b) (c,b,a) n

n n ,

: n 1 2 3 n= , n = n!


[7]

6

;

7 , , , , , , . 7 , 7!=1234567=5040.

7 , 7 .

6 , 6 .

5 , 4 , 3 , 2 1 ( ). ,

7654321=5040.

7

14 .

) . ) 14 : 5 , 4 , 3 2 , , .

) 14 , 14 , 14! .

) 4 4 , 4!=24 .

5 5!=120 , 4 4!=24 , 3 3!=6 2 2!=2 .

14 , , 241202462=829.440 , .


[8]

7) ) . ) . . :

:

:

8) ) 6 6 , 3 ,, .

) , .

9) 1 , 2 , 4 7 , .

.

10) 180 180 . .

4) n . n k (k n) k .

, { } a,b,c= , 3 2 , :

{ } { } { } { } { } { }a,b , b,a , a,c , c,a , b,c , c, b

n k .

n k nk

: nkn!

(n k)!= , k n

n=k , :


[9]

n

n n

n! n! n! n! M(n n)! 0! 1= = = = =

n n n

.

8

15 . 3 .

15 3.

15315! 15! 12! 13 14 15

13 14 15 2730(15 3)! 12! 12!

= = = = =

15 , 15 .

14 , 13 . , 151413=2730

9

1 , 2 , 3 , 4 5. . :

) : ) : 1

() . 3 5 .

535! 5!

() 3 4 5 60(5 3)! 2!= = = = =

) 1 2 3 .


[10]

3 , 2 ( 2 4). 1 2 4

2 , 424! 4!

12(4 2)! 2!= = =

,

()=212=24

() 24 2

()() 60 5= = =

) 1. 2 31

2 3 2 4

{ }2,3,4,5 424! 4!

12(4 2)! 2!= = = .

()=12 , () 12 1()() 60 5= = =

11) ) 0 , 1 , 2 , 3 , 4 5;

) .

i) A: ii) :

12) 7 3 4 . 3 . , .

) : ) :

13) 5 ;


[11]

5.

n k k n , k .

n k nk : n kk n=

10

1 , 2 , 3 , 4 5

3 { } 1,2,3,4,5= , 3 . 5 3 ,

5 33 5 125= =

5 ( ) , , , 555=125

11

20 . ( 365 ).

:

:

.

,

365 20 , 365 2020() 365= =


[12]

365 20 , 36520365! 365!

( ) (365 20)! 345!= = =

20 20

365!( ) 365!345!()() 365 345!365= = = 20

365!() 1 ( ) 1

345!365= =

( 0,41 , 41% , 20 , 2 . 40 90% , 75 100%)

14) 3 , 3 .

15) 10 : .

.

16) 4 7 . ( 1 ). .

6.

n k k .

n k n k

n k

, ,

n k n

k

n n!k k!(n k)!

=


[13]

12

15 ;

4 , 15

4 , : 15 15! 15! 11! 12 13 14 15 13654 4!(15 4)! 4!11! 24 11!

= = = =

13

9 3 , 1 4 , 2 3 3 2 . .

1 9

4 9 9! 9! 1264 4!(9 4)! 4!5!

= = =

5 3 2 .

5 5! 5! 103 3!(5 3)! 3!2!

= = =

.

2 3 , .

, , 3 126101=1260 .

14

10 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 10. 4 .

) : 1 ) : 2

() .


[14]

4 10

10 4 , 10 10! 10!

() 2104 4!(10 4)! 4!6!

= = = =

) () , :

1 , 1 .

3 9 9

3 , 9 9! 9! 843 3!(9 3)! 3!6!

= = =

()=184=84 () 84 2()() 210 5= = =

) , () .

2 , 4 9 9 4.

9 9! 9!

() 1264 4!(9 4)! 4!5!

= = = =

() 126 3

()() 210 5= = =

15

(6 49) 4 .

: 4

() . ,

49 6 , 49 49! 49! 43! 44 45 46 47 48 49

() 139838166 6!(49 6)! 6!43! 6!43!

= = = = =

:

4 6

6 6 6! 154 4!2!

= =

.


[15]

2 43

43 43! 41! 42 43 9032 2!41! 2!41!

= = =

.

, () ()=15903=13545

() 13545

() 0,000969() 13983816= = =

17) 12 14 . )

) : i) 3 ii) 3 iii) 3 ) 6 .

18) 20 , 4 . 3 .

) : 3 ) : ) :

19) 10 (6 49) , :

) 6 ) 5 ) 4


[16]

20) (6 49) ,

) : ) :

21) (6 49) 5 ;

7) ) k n , .

.

.

.

. ,

; .

k n ( ) n . k

; nkn!

(n k)!= ( k n )

, 10 4

1 10 ( 10 ) 2 9 ( 9 )


[17]

3 8 4 7

78910 , !6!10

.

k n ( ) n . k

; n n!k k!(n k)!

=

( k n )

2 , , , , .

25

=

!3!2!5

= 102154=

, 10 , , , , , , , , , . , .

k n

, .

n kk n=

10 4, .

1 10 ( 10 ) 2 10 ( ) 3 10

4 10

10101010 , 410 .

) .


[18]

8.

16

100 1 , 2 , 3 ,, 100, 10 , ( ,

, ,

..). , 5.

() .

, 100

10 , 10010100! 100!

() (100 10)! 90!= = =

: 5

: 5

() 99 ( 5) 10 , 9910

99! 99!( ) (99 10)! 89!= = =

99!() 99!90! 99!89!90 9089!() 0,9100!() 89!100! 89!99!100 100

90!

= = = = = =

() 1 () 1 0,9 0,1= = =

17

100 1 , 2 , 3 ,, 100, 10 , ( , ,

,

..). , 5.


[19]

, , .

() 100 10 , 10() 100=

: 5

: 5

() 99 ( 5) 10 , 10( ) 99=

1010

10( ) 99 99

( )() 100 100

= = =

1099() 1 ( ) 1

100 = =

18

5 , 4 3 , 3 ( ). ) : ) :

() .

5+4+3=12 3 ( ). 12 3 ,

12 12! 12!() 220

3 3!(12 3)! 3!9!

= = = =

) , 3 3 3 .

5 5! 103 3!2!

= =

4 4! 43 3!1!

= =


[20]

3

13

=

.

() ()=10+4+1=15 ( )

() 15 3

()() 220 44= = =

) , 1 , 1 1 .

5

51

=

4

41

=

3

31

=

, , () ()=543=60 ( )

() 60 3

()() 220 11= = =

19

12 4 .

1 12 12! 12! 2203 3!(12 3)! 3!9!

= = =

2 9 9! 9! 843 3!(9 3)! 3!6!

= = =

3 6 6! 6! 203 3!(6 3)! 3!3!

= = =

4 3

13

=

.


[21]

, ,

( ). ,

4!=24 4 .

220 84 20 1 15400

24

=

20

3 10 .

2 .

() . 13 3 ( ) 13 3 ,

13 13! 13! 2863 3!(13 3)! 3!10!

= = =

10 10! 10! 1203 3!(10 3)! 3!7!

= = =

()=286-120=166

: 2

:

2 3

2 1 3 10

3 10 302 1

= =

3 3

13

=

()=30+1=31 ( ) () 31

()() 166= =


[22]

9.

22) 20 10 ( ) 7 . , ,

) ) )

23) 5 1 , 2 , 3 , 4 , 5 7 , , , , , , , .

.

24) 5 1 , 2 , 3 , 4 , 5 7 , , , , , , , 5 . .

25) ) 5 1 , 2 , 3 , 4 , 5 5 , , , , , .

.

26) , , 7 , 2 3 . .

27) 10 , 6 4 ;

28) 12 4 ;

29) 20 , , 5 . 3 . .


[23]

30) 52 6. 2 4 . ( )

31) 5 0 , 1 , 2 , , 9. .

32) , : ) )

33) 10 1 , 2 , . , 10 . ) ) , 12 . 10

34) , , , , , .

.

) : ) :

35) 3 0,5 , 5 1 10 2. 3 .

) : 3 2 ) : 1 0,5 2 1 ) : 3

36) 3 , 2 3 .

.


[24]

37) 6 3 . 3 .

:

38) 6 . :

39) 14 . 9 5 .

) . :

) 2 .

:

:

:

40) ) 7 7 ; ) 7 3 ;

41) 120 . .

42) 25 3 . ,

) )

43) ) 10 , ;

) . 190 , ;


[25]

44) ) ; ) , ;

45) 52 . 3 2 .

46) 7 52 , 3 4 .

47) 30 20 10 . 5 .

) : ) : 2

48) 7 10 .

) 2 ) ) 4

49) 8 3 .

) 3 ) 2 2 1 .

50) 200 20 . 10 . 10 , : ) )

ΣΥΝΔΥΑΣΤΙΚΗ

Documents

Transcript of ΣΥΝΔΥΑΣΤΙΚΗ