Combinatoria resueltos PDF

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MATEMÁTICAS TIMONMATE EJERCICIOS RESUELTOS DE COMBINATORIA Juan Jesús Pascual COMBINATORIA A. Introducción teórica A.1. Variaciones. A.2. Permutaciones. A.3. Combinaciones. A.4. Propiedades de los números combinatorios. A.5. Binomio de Newton. B. Ejercicios resueltos B.1. Cálculo del valor de expres...

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COMBINATORIA A. Introducción teórica

B. Ejercicios resueltos !

"

#

$

A. INTRODUCCIÓN TEÓRICA

=

( − )( − )⋅⋅⋅ =

( − )( − )⋅⋅⋅ ( − + ) = =

! " #

$ #

" " =

=

" "

&

⋅ ⋅ ⋅

( − )( − )⋅⋅⋅ % ⋅ ⋅ =

( − )



%$&'() ( &$

% ' $

 =  

'

  = 

( − )

( "

=

"

)

     =  * 

 = 

  

  − 

   =   

 −   −

  +   

−    =   

  

+ ,

- .

 

( + ) =   *  +   −

  

−

*

 +  

 +    

 

−

 +  

 

−

  +    % 

−%

%

+ +

*

B. EJERCICOS RESUELTOS B.1. Halla el valor de cada expresión:

/

0 1

,

$

$

=

( − )( − )⋅⋅⋅ =

( − )( − )⋅⋅⋅ ( − + ) =

( − )

&'() ( &$



%$= / ⋅ (/ − ) = %*

/

2

3

/

=

/ / ⋅ +⋅ ( ⋅ %⋅ = (⋅ %⋅ (/ − )

= %*

!/ 0 1 ,

$

$

!

$

$

" =

$

$

'

=

! / = / = %/ "( 0 1

"( = ( = ( ⋅ % ⋅ ⋅ = (

,

'/ 0 1

 =  

 = 

( − )

=

"

,  / ' / =   =  

'/ =

/

/ / ⋅ + ⋅( ⋅ % ⋅ ⋅ = ⋅ ⋅( ⋅ % ⋅ ⋅ (/ − )

=

"

/⋅ +

= +

2

= +

+

− '+ %

+

− ' + % = + ⋅ (+ − ) −

0

'( !/

0

+ + ⋅ ⋅% ⋅ ⋅ = +⋅ ( − % (+ − %) %⋅ ⋅

'( = !/

( ( ⋅ %⋅ (( − ) ( = / /⋅ /

= *

=

/



%$&'() ( &$

(

(

0

"(

"(

=

(⋅ % = (⋅ % ⋅

B.2. Halla el valor de x:

!4 −

4

=5

0

4 − 4 (4 − ) = 5 ⇒ 4 − 4 + 4 = 5 ⇒ 4 = 5        +   =    %   4   % 

0 1

  

$

−   − +    −

,        +   =   ⇒ 4 = % − ⇒ 4 = % 4 %

 %6   %6   =  + + 4  4 − 

0 1

  

$

,

= %6 ++ 4 = 4− =

(4 − ) 4 0

  ⇒ 4 = 6  − 

=

   =   

  − 

   =   

  

&'() ( &$



%$(4 − )

4 = ( = * ⇒  4 = −%

= ⇒ 4 −4−

4 (4 − ) (4 − )

B.3. Desarrolla los siguientes productos mediante el binomio de Newton

(4 + ) =   =   4 *

=

 +  

*

 4 

⋅4 +

* ⋅ ( − *)

 +  

 4 * 

⋅( − )

=

⋅4 +

* ⋅ ( − *)

⋅

= 4 + ⋅4 +

%

(4 − ) = % % =  4 + (− ) =   4 % (− * = 4 % − %4 (

(4 + )

+ %4

−

%

 %

 %

) +   4 (− ) +   4 (− ) +   4 * (− ) =     % *

%

%

=

 ( =   4 ( ( *

 (  (  ( +   4 % ( ) +   4 ( ) +   4 (      % ⋅ (⋅ % ⋅ 4% + 4 + ⋅ (⋅ ⋅ 4 + ( = ⋅ ⋅ 4 % + /4 + 5 ⋅ ⋅ 4 + ( *

)

= 4( + ( ⋅

= 4( + ( ⋅

%

)

 ( +   4 * (  (

(

)

=

+

   −  =   + +   =    * 

  =  + −   + +   %

 =  

+

 

%   +  +    (

 − 

 − + ⋅ 

%

⋅ (

+

+

 

 − 

 − 

*  +  +      (   +  +     +

+ ⋅ ⋅ (+ − )

%

+

(

 

*

 

 − 

  +  +     

 − 

  = 

%

  

 − 

  + 

+

% + + ⋅ + ⋅ % ⋅ ( + − %) ( ( ⋅ (+ − ()

(

 −  % 

+

  = 



%$ =  

&'() ( &$

+

 − +  /

%

+

+ (

+ + ⋅

%

+ + ⋅

(

 −  

%

+

  

B.4. Problemas 7' 8 +

$

$

#

9

0 ,

:

$

;

$

;

%

#

,

$

#

:

$

$

$

" ,

'

' ! % + = %+ = (%

7< **

8

#

# $

5

#

9

0 ,

" 3$ =

5

;

>$

>

-

, 5%

= 5 (5 − )(5 − ) = 5 ⋅ ? ⋅ / = %%/

,

$ # #

% 9

,

:

#

7'

*

5 #

&'() ( &$



%$0 , ,

5 %

 5 5 5⋅ ? ⋅ / ⋅ + = +/ ' 5 % =   = = % % (5 − %) % ⋅+

7< 8 # @A -9 7' 8

B 8

9

0 =

(

8

#

- ; 8 (

0 >

; (

(

(

"( = ( = ( ⋅ % ⋅ ⋅ = ( "

"% = % = /

A C C C

"% = % = /

C C C

,

$

B

7' 8

)

#

3

%

(+% 9

0 , #

# 8 3

;

, 3$

D

,

"?

&

? ⋅ ⋅ ⋅ ⋅

&

? ⋅/⋅+⋅(⋅%⋅ & ⋅

7' 8

; ' ={

#} 9 ,

+

$

/*



%$&'() ( &$

0 =

$

;

 / ' / =   =   ,

)

/ / ⋅ +⋅ ( ⋅ %⋅ ⋅ = + = ⋅( ⋅ ⋅ ( ⋅ %⋅ ⋅

/ = (/ − )

+ # 7' E 7' 7' 7' 7'

#

#

8 =,E = ' 09 8 B 8 8 B 8 8 B 8 8 8

# # ;

E9 9 8

8

9

9

0 E

=

= '

,

0

3$

"%

&

%⋅ ⋅ ⋅ ⋅ ⋅ ⋅

F

⋅ * ⋅6 ⋅5⋅? ⋅/ ⋅ +⋅ ( ⋅ % & //% ** % ⋅ ⋅

&

E

$

*

$

=>

" %*

8

,

&

* %⋅ ⋅

&

* ⋅6 ⋅5⋅? ⋅/ ⋅ +⋅ ( ⋅ % ⋅ %⋅ ⋅ ⋅

F * >

"*

;

#

& + ** $

$ ,

&

*

&

⋅ ⋅ ⋅ ⋅ ⋅ ⋅

,

;

#

8

* ⋅6 ⋅5⋅? ⋅/ ⋅ +⋅ ( ⋅ %⋅ ⋅ ⋅

E

& (+%/**

=

&'() ( &$



%$F

$ *

$

>

"6

6

&

⋅ ⋅ ⋅ ⋅ ⋅ ⋅

G

&

=

6⋅5⋅? ⋅/⋅+⋅ ( ⋅%⋅ & 6?* * ⋅ ⋅ #

3;

$

;

,

8

6#

"

' 6 ⋅ "5

8

,

#

"5

;

# E

$

5 ⋅

&

&

5⋅? ⋅/ ⋅ +⋅ ( ⋅ %⋅ & **5* ⋅ ⋅ $

= 6*? *

# $

;

$ 8

#

* E

= >

,

"6 '

6

&

⋅ ⋅ ⋅ ⋅ ⋅ ⋅

) $ / )

$

&

6⋅5⋅? ⋅/⋅+⋅ ( ⋅%⋅ & 6?* * ⋅ ⋅ 3

"

$

$

>1 (6

0 1 (6

$ # (6

)

# $

, /

 (6 (6 = %65%5 / ' (6 / =   =  /  / ( (6 − /)

/ )...