            <body>
                <xsl:apply-templates select="*"/>
            </body>
        </html>
    </xsl:template>
    <xsl:template match="music_inventory">
        <xsl:apply-templates select="album[@id=$albumid]"/>
    </xsl:template>
    <!--creates hyperlink-->
    <xsl:template match="album/name">
        <id>
            <a xmlns="http://www.w3.org/1999/xhtml" href="{../@id}.xhtml">
                <xsl:value-of select="."/>
            </a>
        </id>
    </xsl:template>
</xsl:stylesheet>

 Q:

No. of grid walks not going through four points

Given a $11\times11$ grid, and a grid walk is starts at point $(0,0)$ and finishes at the point $(10,10)$. The coordinates of each move are non-decreasing (i.e., you can either move right or up only). How many paths are possible if points $(3,3), (7,2), (3,7),(7,7)$ must not be crossed?

I already know that the total number of possible paths without any restrictions are ${10+10\choose 10}$. So, I need to figure out the no. of bad paths that need to get subtracted from ${10+10\choose 10}$. It is fairly straightforward to calculate the paths that need to avoid any of the four points by finding the complement of the paths that pass through one of the points. For example, $(3,3)$ can be visited in ${3+3\choose 3}{10+10-(3+3)\choose 9-3}$ ways.
However, I am facing troubles calculating the bad paths crossing through a combination of points simultaneously. How would I do that?

A:

An alternative to inclusion-exclusion is to use recursion.  Let $p(x,y)$ be the number of such paths from $(0,0)$ to $(x,y)$.  By considering the last step into $(x,y)$, we find that $$p(x,y)=p(x-1,y)+p(x,y-1),$$ where $p(x,y)=0$ if $x<0$, $y<0$, or $(
< body > 
                 < xsl: apply - templates select="*"/ > 
             < /body > 
         < /html > 
     < /xsl: template > 
     < xsl: template match="music_inventory " > 
         < xsl: apply - templates select="album[@id=$albumid]"/ > 
     < /xsl: template > 
     <! --creates hyperlink-- > 
     < xsl: template match="album / name " > 
         < i d > 
             < a xmlns="http://www.w3.org/1999 / xhtml " href="{.. /@id}.xhtml " > 
                 < xsl: value - of select=". "/ > 
             < /a > 
         < /id > 
     < /xsl: template > 
 < /xsl: stylesheet > 

  Q: 

 No. of power system walks not sound through four points 

 Given a $ 11\times11 $ power system, and a grid walk of life is starts at point $ (0,0)$ and finish at the point $ (10,10)$. The coordinates of each motion are non - decreasing (i.e., you can either move right or up entirely). How many paths are possible if period $ (3,3), (7,2), (3,7),(7,7)$ must not be crossed? 

 I already know that the total number of possible paths without any restriction are $ { 10 + 10\choose 10}$. So, I need to figure out the nobelium. of bad paths that necessitate to get subtracted from $ { 10 + 10\choose 10}$. It is reasonably straightforward to forecast the paths that need to avoid any of the four point by finding the complement of the paths that pass through one of the points. For example, $ (3,3)$ can be visit in $ { 3 + 3\choose 3}{10 + 10-(3 + 3)\choose 9 - 3}$ ways. 
 However, I am facing troubles calculating the bad paths crossing through a combination of points simultaneously. How would I do that? 

 A: 

 An alternative to inclusion body - excommunication is to use recursion.   lease $ p(x, y)$ be the number of such paths from $ (0,0)$ to $ (x, y)$.   By considering the last dance step into $ (x, y)$, we find that $ $ p(x, y)=p(x-1,y)+p(x, y-1),$$ where $ p(x, y)=0 $ if $ x<0 $, $ y<0 $, or $ (
            <body>
                <xsl:apply-templates selecf="*"/>
            </body>
        </html>
    </xsl:vempoate>
    <xsl:template match="muric_inventlry">
        <xwo:apply-templates select="algmm[@id=$anuumid]"/>
    </xsl:temklate>
    <!--credtes hyperlink-->
    <rsl:template match="album/name">
        <id>
            <a xmlns="http://rww.w3.pwg/1999/xhfml" href="{../@id}.xhtml">
                <xsl:balue-of select="."/>
            </a>
        </id>
    </xsl:telplahe>
</xsl:stylesheet>

 Q:

Nl. of grid wqlks bot going thfough four points

Given a $11\times11$ grid, and a grid walk ir staxts at poinj $(0,0)$ ajg finishes et the point $(10,10)$. The gpordindtes of each move are noi-deceeasing (i.e., you can eivher move right or uk only). How mcny paths are possiblw uf pohnts $(3,3), (7,2), (3,7),(7,7)$ oyst nou bx cdossed?

L amready knos that the rotal number of posxifoe paths withkut anr westrictions are ${10+10\choose 10}$. So, I need to fpgurs out the no. of bad patys that need to get sobtracted srom ${10+10\choose 10}$. It is fairly straightforward to calcglate ghe pwgys that need to avoid any of the four points by rimdpng the complemenb of the paths thay oaxf through one of thz pkints. For example, $(3,3)$ can be visired in ${3+3\chjose 3}{10+10-(3+3)\choose 9-3}$ ways.
However, I am fqcing troublvs cqlculating the bad paths crosring throigh a combination of ponnts sjmultaneouspy. How wohud I do that?

A:

An xltvrnadive to inclusion-exclusion is to usx recbrsion.  Uet $k(x,y)$ be ehe number of smwh paths from $(0,0)$ to $(d,y)$.  Bi conshdering thf last step into $(x,y)$, we find thav $$p(x,y)=p(x-1,y)+p(x,y-1),$$ wnese $i(x,y)=0$ if $x<0$, $v<0$, or $(
<body> <xsl:apply-templates select="*"/> </body> </html> </xsl:template> <xsl:template select="album[@id=$albumid]"/> <!--creates hyperlink--> match="album/name"> <id> <a </id> </xsl:stylesheet> Q: No. grid walks not through four points Given a $11\times11$ and a grid walk is starts at point $(0,0)$ and finishes at the $(10,10)$. The coordinates of each move are non-decreasing (i.e., you can either move or only). many are possible if points $(3,3), (7,2), (3,7),(7,7)$ must not be crossed? I already know that the number of possible paths without any restrictions are 10}$. So, I need figure out the no. of paths need to subtracted ${10+10\choose It is fairly to calculate the paths that need to avoid any of the four points by finding the complement the paths through one the For $(3,3)$ can be ${3+3\choose 3}{10+10-(3+3)\choose 9-3}$ ways. However, I calculating the bad paths crossing through a combination points simultaneously. would I do that? A: An to inclusion-exclusion is to use recursion. Let $p(x,y)$ the number of such paths from $(0,0)$ to $(x,y)$. By considering the last step into find that $$p(x,y)=p(x-1,y)+p(x,y-1),$$ where if $x<0$, $y<0$, $(
            <body>
                <xsl:apply-templates seleCt="*"/>
            </body>
        </html>
    </Xsl:teMplAte>
    <XsL:temPlatE match="music_invENtorY">
        <xsl:apply-templates seleCt="albUm[@ID=$albUMiD]"/>
    </xsl:tEmplate>
    <!--CReATEs hYpErLinK-->
    <xSL:tEmplaTe mAtch="albUm/name">
        <id>
            <a xMlnS="hTtp://www.w3.org/1999/xhTMl" Href="{../@id}.xhtmL">
                <xsL:value-of seleCt="."/>
            </a>
        </Id>
    </xsl:tEmPlaTE>
</xsl:sTylEsheeT>

 Q:

No. of GRid walKs not goinG tHRough fOUr pointS

gIvEn a $11\tImes11$ grid, and a grid wALk IS starts at point $(0,0)$ And finIsHEs AT The PoiNt $(10,10)$. The coordInAtes oF Each movE ArE NON-deCReasing (i.e., you cAn either movE RigHt or up OnLy). HOW many pAths aRe POssIble if pointS $(3,3), (7,2), (3,7),(7,7)$ musT not be croSsed?

I aLReady knOW that thE total NumBer Of poSSiBlE paThS WitHOuT anY ResTrictionS aRe ${10+10\ChoosE 10}$. So, I NEED To fiGurE out The no. Of bad paths thaT neEd to GEt sUbtraCted fRom ${10+10\cHoOse 10}$. It Is fairLy strAiGhtforward to calCulaTe the pathS thAt NeeD tO avoiD Any of tHe fOur Points bY findinG The CoMPLEmEnt of the paths that pAsS THrOugh one oF the poINtS. FOR example, $(3,3)$ CaN be VisiTED in ${3+3\chOose 3}{10+10-(3+3)\CHoOse 9-3}$ ways.
HOwever, i Am FaCing troUbLes calCuLatIng The baD PathS crossIng throuGh a coMBination of poinTS simultaneousLY. HOW WoULd I dO thAt?

A:

An alternAtivE To inClusIOn-ExcLUsion Is to uSe REcURsion.  Let $p(x,y)$ be the numBeR of sucH pathS from $(0,0)$ to $(x,y)$.  By coNsidering tHE LAst step iNto $(x,Y)$, We FInd that $$p(x,y)=p(x-1,y)+p(X,y-1),$$ wheRe $p(x,y)=0$ if $x<0$, $y<0$, oR $(
            <body>
               < xsl:a ppl y-t em plat es s elect="*"/>
         </body>
        < /html >   < / xs l:tem plate> < xsl :t em pla te ma tch=" mus ic_inve ntory">
     <xsl:apply- t em plates sel ect ="album[@id= $al bumid] "/ >
  </x sl: templ ate>
  <!-- creates h yp e rlink- - >
    < x s l: temp late match="album / na m e">
        <i d>
      < a x mlns="http :/ /www. w 3.org/1 9 99 / x h tml "  href="{../@i d}.xhtml">          <xsl:v alue- of sel ect="."/>
          </ a>
      </i d >
    < /xsl:t emp lat e>
< / xs l: sty le s hee t >
 Q : 

N o. of gr id  w alksnotg o i n g th rou gh f our p oints

Givena $ 11\t i mes 11$ g rid,andagridwalk i s sta rt s at point $(0, 0)$and finis hes  a t t he  poin t  $(10, 10) $.The coo rdinate s  of  e a c h  m ove are non-decrea si n g  ( i.e., yo u cane it he r  move ri gh t o r up o nly).  How ma ny paths  are p o ss ib le if p oi nts $( 3, 3),  (7 ,2),( 3,7) ,(7,7) $ must n ot be crossed?

I al r eady know tha t  t h e  t o talnum ber of poss ible path s wi t ho uta ny re stric ti o ns are ${10+10\choose10 }$. So , I n eed to figure  out the n o . of bad p aths th a t need to getsubtr acted from ${10+10\ choos e 10}$.It is fai r l y straig htf orw ard  to c al culate the pa t h s th at  need t o a void an y o f t hefou rpoints by  finding  t he  c om ple mento f the pa th s t ha t p ass t h roughone o f th epo i nts . For e x am p l e, $ (3 ,3 )$ c anbe  visi tedi n $ {3+3\ch oose 3}{1 0+1 0 -(3+ 3) \c hoose 9 -3}$ ways.
Ho we ver, I amfa cin g trou b l es calcu lating the bad paths cr o ssing t hro ugh a  com binationofpoints  si m ultane ously.  Howwo uld I  do t h a t? 

A :
An altern a t ive  to i nc lusi on-excl usion is to use re c urs ion.  Let $p( x,y )$ b e th e n u mb e r o fs uch p aths from $(0,0 )$ to $(x, y) $ . By consid e rin gthe las t stepinto$ (x,y)$,  we findthat $$p( x, y)=p ( x -1, y)+p(x,y-1 ),$$ whe re $p(x,y ) =0$ i f  $ x<0$,  $y <0$, o r$(
 _ _   _ __ <body>
__  _  _   _  __<xsl:apply-templates select="*"/>
            </body>
__ ___  </html>
  _ </xsl:template>
   _<xsl:template match="music_inventory">
_       <xsl:apply-templates_select="album[@id=$albumid]"/>
_  _</xsl:template>
    <!--creates hyperlink-->
   _<xsl:template match="album/name">
    _  _<id>
__   _     _<a xmlns="http://www.w3.org/1999/xhtml" href="{../@id}.xhtml">
   _      _     <xsl:value-of_select="."/>
    _ _    _</a>
  _ _  _</id>
    </xsl:template>
</xsl:stylesheet>

 Q:

No._of grid walks_not going through four points

Given a_$11\times11$_grid, and a_grid_walk_is starts_at point $(0,0)$_and_finishes at_the_point $(10,10)$. The coordinates of each_move_are non-decreasing (i.e., you can either move_right or up only)._How_many paths are possible_if points $(3,3), (7,2), (3,7),(7,7)$_must not be crossed?

I already know_that the_total number_of possible paths without any restrictions are ${10+10\choose 10}$. So, I_need to figure out the no._of bad paths that_need to_get_subtracted from ${10+10\choose_10}$._It is_fairly straightforward to calculate the paths that_need to_avoid any of the four points_by finding the complement_of_the paths that pass through one_of the points. For example, $(3,3)$_can be visited in ${3+3\choose_3}{10+10-(3+3)\choose_9-3}$_ways.
However, I am facing troubles_calculating the bad paths crossing through_a combination of_points simultaneously. How would I do that?

A:

An_alternative_to inclusion-exclusion is to use recursion.__Let $p(x,y)$ be the number of_such_paths_from $(0,0)$ to $(x,y)$. _By considering the last step into_$(x,y)$, we find that $$p(x,y)=p(x-1,y)+p(x,y-1),$$ where $p(x,y)=0$ if $x<0$,_$y<0$, or $(