text: # Number of positive integral solutions to equation x+y+z+w=10 (x>=0,y>=0,z>=2,w>=2) isPlZZZZZZZZZZ answer soon.
  
  jagdish singh singh
  173 Points
  7 years ago
  $\hspace{-0.70 cm}Let z-2=t\geq 0 and w-2=u\geq 0 So eqn.convert into\\\\ x+y+t+2+u+2=10\Rightarrow x+y+z+u=8\;, Where x,y,z,u\geq0\\\\Which is equivalent to arrange 8 stars and 3 bars in row in that way.\\\\\underbrace{*}_{x}|\underbrace{-}_{y}|\underbrace{****}_{z}|\underbrace{***}_{u}, He x=1\;,y=0,z= 4,u=3 is one example\\\\ S Total no.of ways of arranging 8 stars and 3 bars Row is = \binom{8!}{3!\times 5!}$
  jagdish singh singh
  173 Points
  7 years ago
  Sorry I have calculate for non negative Integral solution...................................................
  jagdish singh singh
  173 Points
  7 years ago
  To Admin when i post my answer it did not take My answer and show a message answer already exists.plz help me. anywhy answer is $\displaystyle \binom{11}{3}$
  gandhi1234
  10 Points
  7 years ago
  But the answer given is 84!!!!!!!!!.How????????????????.Plz provide the answer in a complete manner.!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
  
  jagdish singh singh
  173 Points
  7 years ago
  Actually You are Right Answer is =84. Here is my solution.......................................................$\hspace{-.70 cm} Let x+1=x'\geq 1\;,y+1=y'\geq 1 and z-1=z'\geq 1\;,u-1=u'\geq 1\\\\So equation is x'+y'+z'+u'=10\;, Where x',y',z',u'\geq 0\\\\Which is equivalent to arange 3 bars in between 10 stars.\\\\ Like this way \underbrace{***}_{x'}|\underbrace{**}_{y'}|\underbrace{****}_{z'}|\underbrace{*}_{u'}\\\\So Total no. of ways is =\binom{9}{3} = 84.$
  gandhi1234
  10 Points
  7 years ago
  But z>=2 and w>=2 and you take z>=1 and w>=1.How is it possible?My answer is coming out to be 7C3.Is it correct???????????