Combinatoria Clase Martes 29 Completo

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Dernuestl€

eue, Cl.

pba,r n-E+ V

Cl,

=ry

La

n-k+j

L

t4

(n-r+¿)

I

*

L(^- k+ L)

I

l kü rJ.

{n I )

kü*"sr]{^-olñ NJ

-*-'..-....***-_

k (n*B,t

tuu I

*r

vl (n-d

J

\_K I

n /'--'n \: r =Ñ ir Dern"estre que

--\".-,^*t¿-

pbo,

n

l.

n_

F_

e";

n-l¿ n

h-r)

I

(n-r.) Cn-'.- t)l a !

ni

n ,, + éLCI 2t C + "t -Ld I.4= Cal.ular

r

So\,

M=Lct+ci (/'-€? **:* +(

f''1 =

2n I

\

zl *

n-L

I

n,2^n\

H= 2"*(e+rn\t) H= tn+z)i-n

n

"+ *

rJ

-*-. " ' +(^+¿)

d"

I*Lt +ec?+sc?*.. .+nc "rJ "

;J

J

6

G,,rh 2 Calculqn:

CT+C? +C3 {,,.+ffi

=St#+c?c *d ^S -r c? + Ci Ct 1Lo". J-C-ffi *

IJ

-f--,+d-l * (\*o f\,1

t¿

L*ü

-A + , Ci C: +... + r9l \-9'6 * r) \* o

e¿ r."-+ q a

r-¡Y --L,;-

j:

C,;: - C: /" nloo L-.+ I

@

G.,ía 2

#*G-)

Sr.'pl;Aco-r:

C'g+C\i+tl+d

+fil

ñ.rnneradori'

i$s€

v\

FS CT = +[g

C

? --üc?*F#*

Di ^m _tr* L_+ L-

I

= C'á Denornirnctur:

'

CT +C',f +Cf ^rF r'J&

+

J

d

Jg .*, tg 'é: '=L( + Cf r--. ..r__J-j6 'i.-L e -? +cT, /-$, ¡ - ro L-6 + e i' *Le ¡ ,,^?o **>---*)* f2o€ \*p + ¡.?0 \__ g

.

\\A

LorY) hurnerador

n2l -¿r

I

y

clenornrnador

Son

iguales,

l*

res p,;est+

es ¿,

@

Gufa

2

r Calc,.rtar (n'.'unuf.

lo eypresidn:

en

c;-r

= SurnandoS Solución:

to /u_g

+c!

.-'29 Lp

+ C'j +.^. + CT =

--t

. _!¿ --r +c'l +Cg Lr *"'$C^* =C:;*L Lo

g't

tr.!

j

g! +(f+...{

r^rz

/-.h \-h-g

= CT-

an^ t-- m-9

= c;

J

\l-1 cY *"-*e ';'+... a r

I +cj,l

-l +cf

el , = ciJ -l +cs

,z ,ln

(-rn-¡o + C^g

r'rzq r__p_

+c: .1 -------.r--, O

/

4

rn+ j. =2g

rn*g

rn+I" = 29

fn_ZB

rn=Z&

,r rn_g=n

= P =) F= ZS*g= ¿g

ñn+n+p

\/

:$ =

zg+J g+Lq /

ZE-g

=zg^p 19 =zj^ p

P=29-lg

p= /0

zo-g= n

no 4g

_rr

*

'J rn-g = Z9_F..,_

,1

I

t¡do

rn-g=

|

Z8r9 =

n

l9=n ffi.rnfp: ?t*lg+to =S* '

* I2esolver lc ecuccidn.: C?=

44d.

y'

.¡/ \ Sol: z^Lzn_jl{z+¿) _r

\

tn>1-.1?

-a
'rl 3]

zn [en

¿)

fun*]

n Len-¿)

3fk^-¿

)(n_¿

? J

"

t

2L2n-t

1

.

22 n [n-_t ]

=

2zn [n-¿)

= Qzn{n*\

.t J

-::- \-,

?n-L #=

,t 4l

_5

2n*

!"* B3

l^)= i"-E-

*

Calc.,lqr

'>¿'eh a: r=x * + = \--bó.r+

l\

I

\l

presenbn Cog

ee$óSl

o-)

1=Zx.1} + ?V=ft{ y= =) lZ h\ ñx -! ¿-x \-+ = L*v_(e*_r+) J=x-

Lzx-t+)

==5 y,-zx = r?- _t +

-X=

.-f O

X= JO

r-' >t

x= 12 =) r^ L2 ''L2 L + = L¿(ra)-r?

5i y= ro:i CT = efo-,. <-\

'i.

¡^ pos ¡eSpuesta$: V= LO

d

V=

{2

df = zC*;' sxtex-¿)(zv-z) ñ -_A

Y Pesolver

k*d@")

rntrn-r)[rn+Ynn-:)

4l '..

bJ-

Y=3

(t-z){rn-s)=

rn[rn-d

z!

{o

2v

66

Y = Zy*g 3 = ?x*X

=?O

fu_z['n_r)

?x * .z[ex-g) X . = 2x-3. 33

' C! = 2acl."

alto

n".2 ru t_¿rn-grn-F6

=ZqC

lnlsnn*6_2r+o=e fnf-srn-2311 I

=

g

frn * ra) [rn + t3) =o =) nn=l &

'234 I'l +

3q

12 13 13

r-a

lu 1l I I

,p"

pgnnostr-ar Que

/^n'L ' J-t -n-! \_rfl

i C:. =

\-t¡¡-2

(n-¿)l

pba: C?;i + d;,

f,n-r)-(.-,ü I ln-

¿)

(n-

tn-¿-m)l tn l

{ro'-r)l

|

+

(n-l )l (n-rn^l)l m('r"¿)J

r",)J (.n-¿)l

{n-¿ ) I

*

(n-^)("-^-r).r h¿)l

(n--t),I

(.---r)l "\

rnr{n -rn IUñ-rnr'-j

(n*r ))

[- hn+

).ltr+ )l ra

n- rnü

mJ [n-nn)J

nl '(^*r., --ñ r_

*

)J rn

.l



^n I D"rnostrar que j Co= r so)r

(*n

-, a

C?=r-$| -

I

n -'

Cnt t

C

nl nJo

=

é --nJ

Ólnl

I 1

1i'

=a* =L

nJ (n-n)i n!

C)

I

\

L

---

1_

=L

r" (*+)l-

'11

¡-. r)

\_rn = (- n-rn , n= nr)

* Der.'' ostrar que: r\ r, \-rn- ñ! tn-nn))rn

l

NJ

rnJ =

(n-^): [-t*r"t]l

f"-tn-rl] l

NJ

* Demostrar Que:

sc*

\*

rn

cT = d'c:-: -t 1

l

nl

tn*.)

:-e-.

Cn-rl!r l

n!f

I

]

¡Ff-rn+r) J(t"+

il

.t ;

rJ tn-r.)J .tt*-'r-)'F i

rl

nJ ml

il

f¡ -t-l.ñ+(.

n 5 (*

n'frn

An ,r, _m_d_

*

/^' ñ

r/-- n- ¿

\- nn = Fn

\-,

rn-.¿

n -^-l \_fh_l na m{m-di {n-r)} NJ rn .! [n_rn)

J

.ln *

Qrennuestre ,\.ti,fl,\-k

gue; '

/..

/1rn+ñ+t

t--

(

y1

k=O

,nrr¡+z 2 (-1 + ru C. nn!+

+Ca* +... +CT

¡-'' ro. I !. I,alñln \*r-)

CT*+ CT* .-1... + dR o o

¡

/,1rn+n-l L

rl t c

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